# -*- coding: utf-8 -*-
r"""
Finite posets

This module implements finite partially ordered sets. It defines:

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    :class:`FinitePoset` | A class for finite posets
    :class:`FinitePosets_n` | A class for finite posets up to isomorphism (i.e. unlabeled posets)
    :meth:`Poset` | Construct a finite poset from various forms of input data.
    :meth:`is_poset` | Return ``True`` if a directed graph is acyclic and transitively reduced.

List of Poset methods
---------------------

**Comparing, intervals and relations**

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    :meth:`~FinitePoset.is_less_than` | Return ``True`` if `x` is strictly less than `y` in the poset.
    :meth:`~FinitePoset.is_greater_than` | Return ``True`` if `x` is strictly greater than `y` in the poset.
    :meth:`~FinitePoset.is_lequal` | Return ``True`` if `x` is less than or equal to `y` in the poset.
    :meth:`~FinitePoset.is_gequal` | Return ``True`` if `x` is greater than or equal to `y` in the poset.
    :meth:`~FinitePoset.compare_elements` | Compare two element of the poset.
    :meth:`~FinitePoset.closed_interval` | Return the list of elements in a closed interval of the poset.
    :meth:`~FinitePoset.open_interval` | Return the list of elements in an open interval of the poset.
    :meth:`~FinitePoset.relations` | Return the list of relations in the poset.
    :meth:`~FinitePoset.relations_iterator` | Return an iterator over relations in the poset.
    :meth:`~FinitePoset.order_filter` | Return the upper set generated by elements.
    :meth:`~FinitePoset.order_ideal` | Return the lower set generated by elements.

**Covering**

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    :meth:`~FinitePoset.covers` | Return ``True`` if ``y`` covers ``x``.
    :meth:`~FinitePoset.lower_covers` | Return elements covered by given element.
    :meth:`~FinitePoset.upper_covers` | Return elements covering given element.
    :meth:`~FinitePoset.cover_relations` | Return the list of cover relations.
    :meth:`~FinitePoset.lower_covers_iterator` | Return an iterator over elements covered by given element.
    :meth:`~FinitePoset.upper_covers_iterator` | Return an iterator over elements covering given element.
    :meth:`~FinitePoset.cover_relations_iterator` | Return an iterator over cover relations of the poset.
    :meth:`~FinitePoset.common_upper_covers` | Return the list of all common upper covers of the given elements.
    :meth:`~FinitePoset.common_lower_covers` | Return the list of all common lower covers of the given elements.
    :meth:`~FinitePoset.meet` | Return the meet of given elements if it exists; ``None`` otherwise.
    :meth:`~FinitePoset.join` | Return the join of given elements if it exists; ``None`` otherwise.

**Properties of the poset**

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    :meth:`~FinitePoset.cardinality` | Return the number of elements in the poset.
    :meth:`~FinitePoset.height` | Return the number of elements in a longest chain of the poset.
    :meth:`~FinitePoset.width` | Return the number of elements in a longest antichain of the poset.
    :meth:`~FinitePoset.relations_number` | Return the number of relations in the poset.
    :meth:`~FinitePoset.dimension` | Return the dimension of the poset.
    :meth:`~FinitePoset.jump_number` | Return the jump number of the poset.
    :meth:`~FinitePoset.has_bottom` | Return ``True`` if the poset has a unique minimal element.
    :meth:`~FinitePoset.has_top` | Return ``True`` if the poset has a unique maximal element.
    :meth:`~FinitePoset.is_bounded` | Return ``True`` if the poset has both unique minimal and unique maximal element.
    :meth:`~FinitePoset.is_chain` | Return ``True`` if the poset is totally ordered.
    :meth:`~FinitePoset.is_connected` | Return ``True`` if the poset is connected.
    :meth:`~FinitePoset.is_graded` | Return ``True`` if all maximal chains of the poset has same length.
    :meth:`~FinitePoset.is_ranked` | Return ``True`` if the poset has a rank function.
    :meth:`~FinitePoset.is_rank_symmetric` | Return ``True`` if the poset is rank symmetric.
    :meth:`~FinitePoset.is_series_parallel` | Return ``True`` if the poset can be built by ordinal sums and disjoint unions.
    :meth:`~FinitePoset.is_greedy` | Return ``True`` if all greedy linear extensions have equal number of jumps.
    :meth:`~FinitePoset.is_jump_critical` | Return ``True`` if removal of any element reduces the jump number.
    :meth:`~FinitePoset.is_eulerian` | Return ``True`` if the poset is Eulerian.
    :meth:`~FinitePoset.is_incomparable_chain_free` | Return ``True`` if the poset is (m+n)-free.
    :meth:`~FinitePoset.is_slender` | Return ``True`` if the poset is slender.
    :meth:`~FinitePoset.is_sperner` | Return ``True`` if the poset is Sperner.
    :meth:`~FinitePoset.is_join_semilattice` | Return ``True`` is the poset has a join operation.
    :meth:`~FinitePoset.is_meet_semilattice` | Return ``True`` if the poset has a meet operation.

**Minimal and maximal elements**

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    :meth:`~FinitePoset.bottom` | Return the bottom element of the poset, if it exists.
    :meth:`~FinitePoset.top` | Return the top element of the poset, if it exists.
    :meth:`~FinitePoset.maximal_elements` | Return the list of the maximal elements of the poset.
    :meth:`~FinitePoset.minimal_elements` | Return the list of the minimal elements of the poset.

**New posets from old ones**

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    :meth:`~FinitePoset.disjoint_union` | Return the disjoint union of the poset with other poset.
    :meth:`~FinitePoset.ordinal_sum` | Return the ordinal sum of the poset with other poset.
    :meth:`~FinitePoset.product` | Return the Cartesian product of the poset with other poset.
    :meth:`~FinitePoset.ordinal_product` | Return the ordinal product of the poset with other poset.
    :meth:`~FinitePoset.rees_product` | Return the Rees product of the poset with other poset.
    :meth:`~FinitePoset.lexicographic_sum` | Return the lexicographic sum of posets.
    :meth:`~FinitePoset.star_product` | Return the star product of the poset with other poset.
    :meth:`~FinitePoset.with_bounds` | Return the poset with bottom and top element adjoined.
    :meth:`~FinitePoset.without_bounds` | Return the poset with bottom and top element removed.
    :meth:`~FinitePoset.dual` | Return the dual of the poset.
    :meth:`~FinitePoset.completion_by_cuts` | Return the Dedekind-MacNeille completion of the poset.
    :meth:`~FinitePoset.intervals_poset` | Return the poset of intervals of the poset.
    :meth:`~FinitePoset.connected_components` | Return the connected components of the poset as subposets.
    :meth:`~FinitePoset.factor` | Return the decomposition of the poset as a Cartesian product.
    :meth:`~FinitePoset.ordinal_summands` | Return the ordinal summands of the poset.
    :meth:`~FinitePoset.subposet` | Return the subposet containing elements with partial order induced by this poset.
    :meth:`~FinitePoset.random_subposet` | Return a random subposet that contains each element with given probability.
    :meth:`~FinitePoset.relabel` | Return a copy of this poset with its elements relabelled.
    :meth:`~FinitePoset.canonical_label` | Return copy of the poset canonically (re)labelled to integers.
    :meth:`~FinitePoset.slant_sum` | Return the slant sum poset of two posets.

**Chains & antichains**

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    :meth:`~FinitePoset.is_chain_of_poset` | Return ``True`` if elements in the given list are comparable.
    :meth:`~FinitePoset.is_antichain_of_poset` | Return ``True`` if elements in the given list are incomparable.
    :meth:`~FinitePoset.chains` | Return the chains of the poset.
    :meth:`~FinitePoset.antichains` | Return the antichains of the poset.
    :meth:`~FinitePoset.maximal_chains` | Return the maximal chains of the poset.
    :meth:`~FinitePoset.maximal_antichains` | Return the maximal antichains of the poset.
    :meth:`~FinitePoset.maximal_chains_iterator` | Return an iterator over the maximal chains of the poset.
    :meth:`~FinitePoset.maximal_chain_length` | Return the maximum length of maximal chains of the poset.
    :meth:`~FinitePoset.antichains_iterator` | Return an iterator over the antichains of the poset.
    :meth:`~FinitePoset.random_maximal_chain` | Return a random maximal chain.
    :meth:`~FinitePoset.random_maximal_antichain` | Return a random maximal antichain.

**Drawing**

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    :meth:`~FinitePoset.show` | Display the Hasse diagram of the poset.
    :meth:`~FinitePoset.plot` | Return a Graphic object corresponding the Hasse diagram of the poset.
    :meth:`~FinitePoset.graphviz_string` | Return a representation in the DOT language, ready to render in graphviz.

**Comparing posets**

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    :meth:`~FinitePoset.is_isomorphic` | Return ``True`` if both posets are isomorphic.
    :meth:`~FinitePoset.is_induced_subposet` | Return ``True`` if given poset is an induced subposet of this poset.

**Polynomials**

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    :meth:`~FinitePoset.chain_polynomial` | Return the chain polynomial of the poset.
    :meth:`~FinitePoset.characteristic_polynomial` | Return the characteristic polynomial of the poset.
    :meth:`~FinitePoset.f_polynomial` | Return the f-polynomial of the poset.
    :meth:`~FinitePoset.flag_f_polynomial` | Return the flag f-polynomial of the poset.
    :meth:`~FinitePoset.h_polynomial` | Return the h-polynomial of the poset.
    :meth:`~FinitePoset.flag_h_polynomial` | Return the flag h-polynomial of the poset.
    :meth:`~FinitePoset.order_polynomial` | Return the order polynomial of the poset.
    :meth:`~FinitePoset.zeta_polynomial` | Return the zeta polynomial of the poset.
    :meth:`~FinitePoset.kazhdan_lusztig_polynomial` | Return the Kazhdan-Lusztig polynomial of the poset.
    :meth:`~FinitePoset.coxeter_polynomial` | Return the characteristic polynomial of the Coxeter transformation.
    :meth:`~FinitePoset.degree_polynomial` | Return the generating polynomial of degrees of vertices in the Hasse diagram.
    :meth:`~FinitePoset.p_partition_enumerator` | Return a `P`-partition enumerator of the poset.

**Polytopes**

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    :meth:`~FinitePoset.chain_polytope` | Return the chain polytope of the poset.
    :meth:`~FinitePoset.order_polytope` | Return the order polytope of the poset.

**Graphs**

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    :meth:`~FinitePoset.hasse_diagram` | Return the Hasse diagram of the poset as a directed graph.
    :meth:`~FinitePoset.cover_relations_graph` | Return the (undirected) graph of cover relations.
    :meth:`~FinitePoset.comparability_graph` | Return the comparability graph of the poset.
    :meth:`~FinitePoset.incomparability_graph` | Return the incomparability graph of the poset.
    :meth:`~FinitePoset.frank_network` | Return Frank's network of the poset.
    :meth:`~FinitePoset.linear_extensions_graph` | Return the linear extensions graph of the poset.

**Linear extensions**

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    :meth:`~FinitePoset.is_linear_extension` | Return ``True`` if the given list is a linear extension of the poset.
    :meth:`~FinitePoset.linear_extension` | Return a linear extension of the poset.
    :meth:`~FinitePoset.linear_extensions` | Return the enumerated set of all the linear extensions of the poset.
    :meth:`~FinitePoset.promotion` | Return the (extended) promotion on the linear extension of the poset.
    :meth:`~FinitePoset.evacuation` | Return evacuation on the linear extension associated to the poset.
    :meth:`~FinitePoset.with_linear_extension` | Return a copy of ``self`` with a different default linear extension.
    :meth:`~FinitePoset.random_linear_extension` | Return a random linear extension.

**Matrices**

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    :meth:`~FinitePoset.lequal_matrix` | Computes the matrix whose ``(i,j)`` entry is 1 if ``self.linear_extension()[i] < self.linear_extension()[j]`` and 0 otherwise.
    :meth:`~FinitePoset.moebius_function` | Return the value of Möbius function of given elements in the poset.
    :meth:`~FinitePoset.moebius_function_matrix` | Return a matrix whose ``(i,j)`` entry is the value of the Möbius function evaluated at ``self.linear_extension()[i]`` and ``self.linear_extension()[j]``.
    :meth:`~FinitePoset.coxeter_transformation` | Return the matrix of the Auslander-Reiten translation acting on the Grothendieck group of the derived category of modules.
    :meth:`~FinitePoset.coxeter_smith_form` | Return the Smith form of the Coxeter transformation.

**Miscellaneous**

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    :meth:`~FinitePoset.sorted` | Return given list sorted by the poset.
    :meth:`~FinitePoset.isomorphic_subposets` | Return all subposets isomorphic to another poset.
    :meth:`~FinitePoset.isomorphic_subposets_iterator` | Return an iterator over the subposets isomorphic to another poset.
    :meth:`~FinitePoset.has_isomorphic_subposet` | Return ``True`` if the poset contains a subposet isomorphic to another poset.
    :meth:`~FinitePoset.list` | List the elements of the poset.
    :meth:`~FinitePoset.cuts` | Return the cuts of the given poset.
    :meth:`~FinitePoset.dilworth_decomposition` | Return a partition of the points into the minimal number of chains.
    :meth:`~FinitePoset.greene_shape` | Computes the Greene-Kleitman partition aka Greene shape of the poset ``self``.
    :meth:`~FinitePoset.incidence_algebra` | Return the incidence algebra of ``self``.
    :meth:`~FinitePoset.is_EL_labelling` | Return whether ``f`` is an EL labelling of the poset.
    :meth:`~FinitePoset.isomorphic_subposets_iterator` | Return an iterator over the subposets isomorphic to another poset.
    :meth:`~FinitePoset.isomorphic_subposets` | Return all subposets isomorphic to another poset.
    :meth:`~FinitePoset.level_sets` | Return elements grouped by maximal number of cover relations from a minimal element.
    :meth:`~FinitePoset.order_complex` | Return the order complex associated to this poset.
    :meth:`~FinitePoset.random_order_ideal` | Return a random order ideal of ``self`` with uniform probability.
    :meth:`~FinitePoset.rank` | Return the rank of an element, or the rank of the poset.
    :meth:`~FinitePoset.rank_function` | Return a rank function of the poset, if it exists.
    :meth:`~FinitePoset.unwrap` | Unwraps an element of this poset.
    :meth:`~FinitePoset.atkinson` | Return the `a`-spectrum of a poset whose undirected Hasse diagram is a forest.
    :meth:`~FinitePoset.spectrum` | Return the `a`-spectrum of this poset.

Classes and functions
---------------------
"""

# ****************************************************************************
#       Copyright (C) 2008 Peter Jipsen <jipsen@chapman.edu>
#       Copyright (C) 2008 Franco Saliola <saliola@gmail.com>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  https://www.gnu.org/licenses/
# ****************************************************************************

from copy import copy, deepcopy
from sage.misc.cachefunc import cached_method
from sage.misc.lazy_attribute import lazy_attribute
from sage.misc.misc_c import prod
from sage.functions.other import floor
from sage.functions.other import binomial
from sage.categories.category import Category
from sage.categories.sets_cat import Sets
from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
from sage.categories.posets import Posets
from sage.categories.finite_posets import FinitePosets
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.parent import Parent
from sage.rings.integer import Integer
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.polynomial.polynomial_ring import polygen
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.graphs.digraph import DiGraph
from sage.graphs.digraph_generators import digraphs
from sage.combinat.posets.hasse_diagram import HasseDiagram
from sage.combinat.posets.elements import PosetElement
from sage.combinat.combinatorial_map import combinatorial_map
from sage.combinat.subset import Subsets
from .linear_extensions import LinearExtensionsOfPoset


def Poset(data=None, element_labels=None, cover_relations=False, linear_extension=False, category=None, facade=None, key=None):
    r"""
    Construct a finite poset from various forms of input data.

    INPUT:

    - ``data`` -- different input are accepted by this constructor:

      1. A two-element list or tuple ``(E, R)``, where ``E`` is a
         collection of elements of the poset and ``R`` is a collection
         of relations ``x <= y``, each represented as a two-element
         list/tuple/iterable such as ``[x, y]``. The poset is then
         the transitive closure of the provided relations. If
         ``cover_relations=True``, then ``R`` is assumed to contain
         exactly the cover relations of the poset. If ``E`` is empty,
         then ``E`` is taken to be the set of elements appearing in
         the relations ``R``.

      2. A two-element list or tuple ``(E, f)``, where ``E`` is the set
         of elements of the poset and ``f`` is a function such that,
         for any pair ``x, y`` of elements of ``E``, ``f(x, y)``
         returns whether ``x <= y``. If ``cover_relations=True``, then
         ``f(x, y)`` should instead return whether ``x`` is covered by
         ``y``.

      3. A dictionary of upper covers: ``data[x]`` is
         a list of the elements that cover the element `x` in the poset.

      4. A list or tuple of upper covers: ``data[x]`` is
         a list of the elements that cover the element `x` in the poset.

         If the set of elements is not a set of consecutive integers
         starting from zero, then:

         - every element must appear in the data, for example in its own entry.
         - ``data`` must be ordered in the same way as sorted elements.

         .. WARNING::

             If data is a list or tuple of length `2`, then it is
             handled by the case 2 above.

      5. An acyclic, loop-free and multi-edge free ``DiGraph``. If
         ``cover_relations`` is ``True``, then the edges of the
         digraph are assumed to correspond to the cover relations of
         the poset. Otherwise, the cover relations are computed.

      6. A previously constructed poset (the poset itself is returned).

    - ``element_labels`` -- (default: ``None``); an optional list or
      dictionary of objects that label the poset elements.

    - ``cover_relations`` -- a boolean (default: ``False``); whether the
      data can be assumed to describe a directed acyclic graph whose
      arrows are cover relations; otherwise, the cover relations are
      first computed.

    - ``linear_extension`` -- a boolean (default: ``False``); whether to
      use the provided list of elements as default linear extension
      for the poset; otherwise a linear extension is computed. If the data
      is given as the pair ``(E, f)``, then ``E`` is taken to be the linear
      extension.

    - ``facade`` -- a boolean or ``None`` (default); whether the
      :meth:`Poset`'s elements should be wrapped to make them aware of the
      Poset they belong to.

      * If ``facade = True``, the :meth:`Poset`'s elements are exactly those
        given as input.

      * If ``facade = False``, the :meth:`Poset`'s elements will become
        :class:`~sage.combinat.posets.posets.PosetElement` objects.

      * If ``facade = None`` (default) the expected behaviour is the behaviour
        of ``facade = True``, unless the opposite can be deduced from the
        context (i.e. for instance if a :meth:`Poset` is built from another
        :meth:`Poset`, itself built with ``facade = False``)

    OUTPUT:

    ``FinitePoset`` -- an instance of the :class:`FinitePoset` class.

    If ``category`` is specified, then the poset is created in this
    category instead of :class:`FinitePosets`.

    .. SEEALSO::

        :class:`Posets`, :class:`~sage.categories.posets.Posets`,
        :class:`FinitePosets`

    EXAMPLES:

    1. Elements and cover relations::

          sage: elms = [1,2,3,4,5,6,7]
          sage: rels = [[1,2],[3,4],[4,5],[2,5]]
          sage: Poset((elms, rels), cover_relations = True, facade = False)
          Finite poset containing 7 elements

       Elements and non-cover relations::

          sage: elms = [1,2,3,4]
          sage: rels = [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]
          sage: P = Poset( [elms,rels] ,cover_relations=False); P
          Finite poset containing 4 elements
          sage: P.cover_relations()
          [[1, 2], [2, 3], [3, 4]]

    2. Elements and function: the standard permutations of [1, 2, 3, 4]
       with the Bruhat order::

          sage: elms = Permutations(4)
          sage: fcn = lambda p,q : p.bruhat_lequal(q)
          sage: Poset((elms, fcn))
          Finite poset containing 24 elements

       With a function that identifies the cover relations: the set
       partitions of `\{1, 2, 3\}` ordered by refinement::

          sage: elms = SetPartitions(3)
          sage: def fcn(A, B):
          ....:     if len(A) != len(B)+1:
          ....:         return False
          ....:     for a in A:
          ....:         if not any(set(a).issubset(b) for b in B):
          ....:             return False
          ....:     return True
          sage: Poset((elms, fcn), cover_relations=True)
          Finite poset containing 5 elements

    3. A dictionary of upper covers::

          sage: Poset({'a':['b','c'], 'b':['d'], 'c':['d'], 'd':[]})
          Finite poset containing 4 elements

    4. A list of upper covers, with range(5) as set of vertices::

          sage: Poset([[1,2],[4],[3],[4],[]])
          Finite poset containing 5 elements

       A list of upper covers, with letters as vertices::

          sage: Poset([["a","b"],["b","c"],["c"]])
          Finite poset containing 3 elements

       A list of upper covers and a dictionary of labels::

          sage: elm_labs = {0:"a",1:"b",2:"c",3:"d",4:"e"}
          sage: P = Poset([[1,2],[4],[3],[4],[]], elm_labs, facade=False)
          sage: P.list()
          [a, b, c, d, e]

       .. WARNING::

         The special case where the argument data is a list or tuple of
         length 2 is handled by the case 2. So you cannot use this
         method to input a 2-element poset.

    5. An acyclic DiGraph.

       ::

          sage: dag = DiGraph({0:[2,3], 1:[3,4], 2:[5], 3:[5], 4:[5]})
          sage: Poset(dag)
          Finite poset containing 6 elements

       Any directed acyclic graph without loops or multiple edges, as long
       as ``cover_relations=False``::

          sage: dig = DiGraph({0:[2,3], 1:[3,4,5], 2:[5], 3:[5], 4:[5]})
          sage: dig.allows_multiple_edges()
          False
          sage: dig.allows_loops()
          False
          sage: dig.transitive_reduction() == dig
          False
          sage: Poset(dig, cover_relations=False)
          Finite poset containing 6 elements
          sage: Poset(dig, cover_relations=True)
          Traceback (most recent call last):
          ...
          ValueError: Hasse diagram is not transitively reduced

    .. rubric:: Default Linear extension

    Every poset `P` obtained with ``Poset`` comes equipped with a
    default linear extension, which is also used for enumerating
    its elements. By default, this linear extension is computed,
    and has no particular significance::

        sage: P = Poset((divisors(12), attrcall("divides")))
        sage: P.list()
        [1, 2, 4, 3, 6, 12]
        sage: P.linear_extension()
        [1, 2, 4, 3, 6, 12]

    You may enforce a specific linear extension using the
    ``linear_extension`` option::

        sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
        sage: P.list()
        [1, 2, 3, 4, 6, 12]
        sage: P.linear_extension()
        [1, 2, 3, 4, 6, 12]

    Depending on popular request, ``Poset`` might eventually get
    modified to always use the provided list of elements as
    default linear extension, when it is one.

    .. SEEALSO:: :meth:`FinitePoset.linear_extensions`

    .. rubric:: Facade posets

    When ``facade = False``, the elements of a poset are wrapped so as to make
    them aware that they belong to that poset::

        sage: P = Poset(DiGraph({'d':['c','b'],'c':['a'],'b':['a']}), facade = False)
        sage: d,c,b,a = list(P)
        sage: a.parent() is P
        True

    This allows for comparing elements according to `P`::

        sage: c < a
        True

    However, this may have surprising effects::

        sage: my_elements = ['a','b','c','d']
        sage: any(x in my_elements for x in P)
        False

    and can be annoying when one wants to manipulate the elements of
    the poset::

        sage: a + b
        Traceback (most recent call last):
        ...
        TypeError: unsupported operand parent(s) for +: 'Finite poset containing 4 elements' and 'Finite poset containing 4 elements'
        sage: a.element + b.element
        'ab'

    By default, facade posets are constructed instead::

        sage: P = Poset(DiGraph({'d':['c','b'],'c':['a'],'b':['a']}))

    In this example, the elements of the poset remain plain strings::

        sage: d,c,b,a = list(P)
        sage: type(a)
        <... 'str'>

    Of course, those strings are not aware of `P`. So to compare two
    such strings, one needs to query `P`::

        sage: a < b
        True
        sage: P.lt(a,b)
        False

    which models the usual mathematical notation `a <_P b`.

    Most operations seem to still work, but at this point there is no
    guarantee whatsoever::

        sage: P.list()
        ['d', 'c', 'b', 'a']
        sage: P.principal_order_ideal('a')
        ['d', 'c', 'b', 'a']
        sage: P.principal_order_ideal('b')
        ['d', 'b']
        sage: P.principal_order_ideal('d')
        ['d']
        sage: TestSuite(P).run()

    .. WARNING::

        :class:`DiGraph` is used to construct the poset, and the
        vertices of a :class:`DiGraph` are converted to plain Python
        :class:`int`'s if they are :class:`Integer`'s::

            sage: G = DiGraph({0:[2,3], 1:[3,4], 2:[5], 3:[5], 4:[5]})
            sage: type(G.vertices()[0])
            <... 'int'>

        This is worked around by systematically converting back the
        vertices of a poset to :class:`Integer`'s if they are
        :class:`int`'s::

            sage: P = Poset((divisors(15), attrcall("divides")), facade = False)
            sage: type(P.an_element().element)
            <... 'sage.rings.integer.Integer'>

            sage: P = Poset((divisors(15), attrcall("divides")), facade=True)
            sage: type(P.an_element())
            <... 'sage.rings.integer.Integer'>

        This may be abusive::

            sage: P = Poset((range(5), operator.le), facade = True)
            sage: P.an_element().parent()
            Integer Ring

    .. rubric:: Unique representation

    As most parents, :class:`Poset` have unique representation (see
    :class:`UniqueRepresentation`). Namely if two posets are created
    from two equal data, then they are not only equal but actually
    identical::

        sage: data1 = [[1,2],[3],[3]]
        sage: data2 = [[1,2],[3],[3]]
        sage: P1 = Poset(data1)
        sage: P2 = Poset(data2)
        sage: P1 == P2
        True
        sage: P1 is P2
        True

    In situations where this behaviour is not desired, one can use the
    ``key`` option::

        sage: P1 = Poset(data1, key = "foo")
        sage: P2 = Poset(data2, key = "bar")
        sage: P1 is P2
        False
        sage: P1 == P2
        False

    ``key`` can be any hashable value and is passed down to
    :class:`UniqueRepresentation`. It is otherwise ignored by the
    poset constructor.

    TESTS::

        sage: P = Poset([[1,2],[3],[3]])
        sage: type(hash(P))
        <... 'int'>

    Bad input::

        sage: Poset([1,2,3], lambda x,y : x<y)
        Traceback (most recent call last):
        ...
        TypeError: element_labels should be a dict or a list if different
        from None. (Did you intend data to be equal to a pair ?)

    Another kind of bad input, digraphs with oriented cycles::

        sage: Poset(DiGraph([[1,2],[2,3],[3,4],[4,1]]))
        Traceback (most recent call last):
        ...
        ValueError: The graph is not directed acyclic

    Some more bad input, an element list with duplicates while
    ``linear_extension=True``::

        sage: Poset(( [1,2,3,3], [[1,2]]), linear_extension=True)
        Traceback (most recent call last):
        ...
        ValueError: the provided list of elements is not a linear extension
        for the poset as it contains duplicate elements
    """
    # Avoiding some errors from the user when data should be a pair
    if not(element_labels is None or isinstance(element_labels, (dict, list))):
        raise TypeError("element_labels should be a dict or a list if "
                        "different from None. (Did you intend data to be "
                        "equal to a pair ?)")

    if isinstance(data, FinitePoset):
        if element_labels is None and category is None and facade is None and linear_extension == data._with_linear_extension:
            return data
        if not linear_extension:
            P = FinitePoset(data, elements=None, category=category, facade=facade)
            if element_labels is not None:
                P = P.relabel(element_labels)
            return P
        else:
            if element_labels is None:
                return FinitePoset(data, elements=data._elements, category=category, facade=facade)
            else:
                return FinitePoset(data, elements=element_labels, category=category, facade=facade)

    # Convert data to a DiGraph
    elements = None
    D = {}
    if data is None:  # type 0
        D = DiGraph()
    elif isinstance(data, DiGraph):  # type 4
        D = deepcopy(data)
    elif isinstance(data, dict):  # type 3: dictionary of upper covers
        D = DiGraph(data, format="dict_of_lists")
    elif isinstance(data, (list, tuple)):  # types 1, 2, 3 (list/tuple)
        if len(data) == 2:  # types 1 or 2
            if callable(data[1]):  # type 2
                elements, function = data
                relations = [[x, y] for x in elements for y in elements
                             if function(x, y)]
            else:  # type 1
                elements, relations = data
                # check that relations are relations
                for r in relations:
                    try:
                        u, v = r
                    except ValueError:
                        raise TypeError("not a list of relations")
            D = DiGraph()
            D.add_vertices(elements)
            D.add_edges(relations, loops=False)
        elif len(data) > 2:
            # type 3, list/tuple of upper covers
            vertices = sorted(set(x for item in data for x in item))
            if len(vertices) != len(data):
                # by default, assuming vertices are the range 0..n
                vertices = range(len(data))
            D = DiGraph({v: [u for u in cov if u != v]
                         for v, cov in zip(vertices, data)},
                        format="dict_of_lists")
        else:
            raise ValueError("not valid poset data")

    # DEBUG: At this point D should be a DiGraph.
    assert isinstance(D, DiGraph), "BUG: D should be a digraph."

    # Determine cover relations, if necessary.
    if not cover_relations:
        from sage.graphs.generic_graph_pyx import transitive_reduction_acyclic
        D = transitive_reduction_acyclic(D)

    # Check that the digraph does not contain loops, multiple edges
    # and is transitively reduced.
    if D.has_loops():
        raise ValueError("Hasse diagram contains loops")
    elif D.has_multiple_edges():
        raise ValueError("Hasse diagram contains multiple edges")
    elif cover_relations and not D.is_transitively_reduced():
        raise ValueError("Hasse diagram is not transitively reduced")

    if element_labels is not None:
        D = D.relabel(element_labels, inplace=False)

    if linear_extension:
        if element_labels is not None:
            elements = element_labels
        elif elements is None:
            # Compute a linear extension of the poset (a topological sort).
            try:
                elements = D.topological_sort()
            except TypeError:
                raise ValueError("Hasse diagram contains cycles")
        # Check for duplicate elements
        elif len(elements) != len(set(elements)):
            raise ValueError("the provided list of elements is not a linear "
                "extension for the poset as it contains duplicate elements")
    else:
        elements = None
    return FinitePoset(D, elements=elements, category=category, facade=facade, key=key)


class FinitePoset(UniqueRepresentation, Parent):
    r"""
    A (finite) `n`-element poset constructed from a directed acyclic graph.

    INPUT:

    - ``hasse_diagram`` -- an instance of
      :class:`~sage.combinat.posets.posets.FinitePoset`, or a
      :class:`DiGraph` that is transitively-reduced, acyclic,
      loop-free, and multiedge-free.

    - ``elements`` -- an optional list of elements, with ``element[i]``
      corresponding to vertex ``i``. If ``elements`` is ``None``, then it is
      set to be the vertex set of the digraph. Note that if this option is set,
      then ``elements`` is considered as a specified linear extension of the poset
      and the `linear_extension` attribute is set.

    - ``category`` -- :class:`FinitePosets`, or a subcategory thereof.

    - ``facade`` -- a boolean or ``None`` (default); whether the
      :class:`~sage.combinat.posets.posets.FinitePoset`'s elements should be
      wrapped to make them aware of the Poset they belong to.

      * If ``facade = True``, the
        :class:`~sage.combinat.posets.posets.FinitePoset`'s elements are exactly
        those given as input.

      * If ``facade = False``, the
        :class:`~sage.combinat.posets.posets.FinitePoset`'s elements will become
        :class:`~sage.combinat.posets.posets.PosetElement` objects.

      * If ``facade = None`` (default) the expected behaviour is the behaviour
        of ``facade = True``, unless the opposite can be deduced from the
        context (i.e. for instance if a
        :class:`~sage.combinat.posets.posets.FinitePoset` is built from another
        :class:`~sage.combinat.posets.posets.FinitePoset`, itself built with
        ``facade = False``)

    - ``key`` -- any hashable value (default: ``None``).

    EXAMPLES::

        sage: uc = [[2,3], [], [1], [1], [1], [3,4]]
        sage: from sage.combinat.posets.posets import FinitePoset
        sage: P = FinitePoset(DiGraph(dict([[i,uc[i]] for i in range(len(uc))])), facade=False); P
        Finite poset containing 6 elements
        sage: P.cover_relations()
        [[5, 4], [5, 3], [4, 1], [0, 2], [0, 3], [2, 1], [3, 1]]
        sage: TestSuite(P).run()
        sage: P.category()
        Category of finite enumerated posets
        sage: P.__class__
        <class 'sage.combinat.posets.posets.FinitePoset_with_category'>

        sage: Q = sage.combinat.posets.posets.FinitePoset(P, facade = False); Q
        Finite poset containing 6 elements

        sage: Q is P
        True

    We keep the same underlying Hasse diagram, but change the elements::

        sage: Q = sage.combinat.posets.posets.FinitePoset(P, elements=[1,2,3,4,5,6], facade=False); Q
        Finite poset containing 6 elements with distinguished linear extension
        sage: Q.cover_relations()
        [[1, 2], [1, 5], [2, 6], [3, 4], [3, 5], [4, 6], [5, 6]]

    We test the facade argument::

        sage: P = Poset(DiGraph({'a':['b'],'b':['c'],'c':['d']}), facade=False)
        sage: P.category()
        Category of finite enumerated posets
        sage: parent(P[0]) is P
        True

        sage: Q = Poset(DiGraph({'a':['b'],'b':['c'],'c':['d']}), facade=True)
        sage: Q.category()
        Category of facade finite enumerated posets
        sage: parent(Q[0]) is str
        True
        sage: TestSuite(Q).run(skip = ['_test_an_element']) # is_parent_of is not yet implemented

    Changing a non facade poset to a facade poset::

        sage: PQ = Poset(P, facade=True)
        sage: PQ.category()
        Category of facade finite enumerated posets
        sage: parent(PQ[0]) is str
        True
        sage: PQ is Q
        True

    Changing a facade poset to a non facade poset::

        sage: QP = Poset(Q, facade = False)
        sage: QP.category()
        Category of finite enumerated posets
        sage: parent(QP[0]) is QP
        True

    Conversion to some other software is possible::

        sage: P = posets.TamariLattice(3)
        sage: libgap(P)  # optional - gap_packages
        <A poset on 5 points>

        sage: P = Poset({1:[2],2:[]})
        sage: macaulay2('needsPackage "Posets"')   # optional - macaulay2
        Posets
        sage: macaulay2(P)   # optional - macaulay2
        Relation Matrix: | 1 1 |
                         | 0 1 |

    .. NOTE::

       A class that inherits from this class needs to define
       ``Element``. This is the class of the elements that the inheriting
       class contains. For example, for this class, ``FinitePoset``,
       ``Element`` is ``PosetElement``.  It can also define ``_dual_class`` which
       is the class of dual posets of this
       class. E.g. ``FiniteMeetSemilattice._dual_class`` is
       ``FiniteJoinSemilattice``.

    TESTS:

    Equality is derived from :class:`UniqueRepresentation`. We check that this
    gives consistent results::

        sage: P = Poset([[1,2],[3],[3]])
        sage: P == P
        True
        sage: Q = Poset([[1,2],[],[1]])
        sage: Q == P
        False
        sage: p1, p2 = Posets(2).list()
        sage: p2 == p1, p1 != p2
        (False, True)
        sage: [[p1.__eq__(p2) for p1 in Posets(2)] for p2 in Posets(2)]
        [[True, False], [False, True]]
        sage: [[p2.__eq__(p1) for p1 in Posets(2)] for p2 in Posets(2)]
        [[True, False], [False, True]]
        sage: [[p2 == p1 for p1 in Posets(3)] for p2 in Posets(3)]
        [[True, False, False, False, False],
         [False, True, False, False, False],
         [False, False, True, False, False],
         [False, False, False, True, False],
         [False, False, False, False, True]]

        sage: [[p1.__ne__(p2) for p1 in Posets(2)] for p2 in Posets(2)]
        [[False, True], [True, False]]
        sage: P = Poset([[1,2,4],[3],[3]])
        sage: Q = Poset([[1,2],[],[1],[4]])
        sage: P != Q
        True
        sage: P != P
        False
        sage: Q != Q
        False
        sage: [[p1.__ne__(p2) for p1 in Posets(2)] for p2 in Posets(2)]
        [[False, True], [True, False]]

        sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
        sage: Q = Poset(P)
        sage: Q == P
        False
        sage: Q = Poset(P, linear_extension=True)
        sage: Q == P
        True
    """
    _lin_ext_type = LinearExtensionsOfPoset
    _desc = 'Finite poset'

    # The parsing of the construction data (like a list of cover relations)
    #   into a :class:`DiGraph` is done in :func:`Poset`.
    @staticmethod
    def __classcall__(cls, hasse_diagram, elements=None, category=None, facade=None, key=None):
        """
        Normalizes the arguments passed to the constructor.

        INPUT:

        - ``hasse_diagram`` -- a :class:`DiGraph` or a :class:`FinitePoset`
          that is labeled by the elements of the poset
        - ``elements`` -- (default: ``None``) the default linear extension
          or ``None`` if no such default linear extension is wanted
        - ``category`` -- (optional) a subcategory of :class:`FinitePosets`
        - ``facade`` -- (optional) boolean if this is a facade parent or not
        - ``key`` -- (optional) a key value

        TESTS::

            sage: P = sage.combinat.posets.posets.FinitePoset(DiGraph())
            sage: type(P)
            <class 'sage.combinat.posets.posets.FinitePoset_with_category'>
            sage: TestSuite(P).run()

        See also the extensive tests in the class documentation.

        We check that :trac:`17059` is fixed::

            sage: p = Poset()
            sage: p is Poset(p, category=p.category())
            True
        """
        assert isinstance(hasse_diagram, (FinitePoset, DiGraph))
        if isinstance(hasse_diagram, FinitePoset):
            poset = hasse_diagram
            if category is None:
                category = poset.category()
            if facade is None:
                facade = poset in Sets().Facade()
            if elements is None:
                relabel = {i: x for i, x in enumerate(poset._elements)}
            else:
                elements = tuple(elements)
                relabel = {i: x for i, x in enumerate(elements)}
            hasse_diagram = poset._hasse_diagram.relabel(relabel, inplace=False)
            hasse_diagram = hasse_diagram.copy(immutable=True)
        else:
            hasse_diagram = HasseDiagram(hasse_diagram, data_structure="static_sparse")
            if facade is None:
                facade = True
            if elements is not None:
                elements = tuple(elements)
        # Standardize the category by letting the Facade axiom be carried
        #   by the facade variable
        if category is not None and category.is_subcategory(Sets().Facade()):
            category = category._without_axiom("Facade")
        category = Category.join([FinitePosets().or_subcategory(category), FiniteEnumeratedSets()])
        return super(FinitePoset, cls).__classcall__(cls, hasse_diagram=hasse_diagram,
                                                     elements=elements,
                                                     category=category, facade=facade,
                                                     key=key)

    def __init__(self, hasse_diagram, elements, category, facade, key):
        r"""
        EXAMPLES::

            sage: P = Poset(DiGraph({'a':['b'],'b':['c'],'c':['d']}), facade = False)
            sage: type(P)
            <class 'sage.combinat.posets.posets.FinitePoset_with_category'>

        The internal data structure currently consists of:

        - the Hasse diagram of the poset, represented by a DiGraph
          with vertices labelled 0,...,n-1 according to a linear
          extension of the poset (that is if `i \mapsto j` is an edge
          then `i<j`), together with some extra methods (see
          :class:`sage.combinat.posets.hasse_diagram.HasseDiagram`)::

            sage: P._hasse_diagram
            Hasse diagram of a poset containing 4 elements
            sage: P._hasse_diagram.cover_relations()
            [(0, 1), (1, 2), (2, 3)]

        - a tuple of the original elements, not wrapped as elements of
          ``self`` (but see also ``P._list``)::

            sage: P._elements
            ('a', 'b', 'c', 'd')

          ``P._elements[i]`` gives the element of ``P`` corresponding
          to the vertex ``i``

        - a dictionary mapping back elements to vertices::

            sage: P._element_to_vertex_dict
            {'a': 0, 'b': 1, 'c': 2, 'd': 3}

        - and a boolean stating whether the poset is a facade poset::

            sage: P._is_facade
            False

        This internal data structure is subject to change at any
        point. Do not break encapsulation!

        TESTS::

            sage: TestSuite(P).run()

        See also the extensive tests in the class documentation.
        """
        Parent.__init__(self, category=category, facade=facade)
        if elements is None:
            self._with_linear_extension = False
            # Compute a linear extension of the poset (a topological sort).
            try:
                elements = tuple(hasse_diagram.topological_sort())
            except TypeError:
                raise ValueError("Hasse diagram contains cycles")
        else:
            self._with_linear_extension = True
        # Work around the fact that, currently, when a DiGraph is
        # created with Integer's as vertices, those vertices are
        # converted to plain int's. This is a bit abusive.
        self._elements = tuple(Integer(i) if isinstance(i, int) else i
                               for i in elements)
        # Relabel using the linear_extension.
        # So range(len(D)) becomes a linear extension of the poset.
        rdict = {self._elements[i]: i for i in range(len(self._elements))}
        self._hasse_diagram = HasseDiagram(hasse_diagram.relabel(rdict, inplace=False), data_structure="static_sparse")
        self._element_to_vertex_dict = {self._elements[i]: i
                                        for i in range(len(self._elements))}
        self._is_facade = facade

    @lazy_attribute
    def _list(self):
        """
        The list of the elements of ``self``, each wrapped to have
        ``self`` as parent

        EXAMPLES::

            sage: P = Poset(DiGraph({'a':['b'],'b':['c'],'c':['d']}), facade = False)
            sage: L = P._list; L
            (a, b, c, d)
            sage: type(L[0])
            <class 'sage.combinat.posets.posets.FinitePoset_with_category.element_class'>
            sage: L[0].parent() is P
            True

        Constructing them once for all makes future conversions
        between the vertex id and the element faster. This also
        ensures unique representation of the elements of this poset,
        which could be used to later speed up certain operations
        (equality test, ...)
        """
        if self._is_facade:
            return self._elements
        else:
            return tuple(self.element_class(self, self._elements[vertex], vertex)
                         for vertex in range(len(self._elements)))

    # This defines the type (class) of elements of poset.
    Element = PosetElement

    def _element_to_vertex(self, element):
        """
        Given an element of the poset (wrapped or not), returns the
        corresponding vertex of the Hasse diagram.

        EXAMPLES::

            sage: P = Poset((divisors(15), attrcall("divides")))
            sage: list(P)
            [1, 3, 5, 15]
            sage: x = P(5)
            sage: P._element_to_vertex(x)
            2

        The same with a non-wrapped element of `P`::

            sage: P._element_to_vertex(5)
            2

        TESTS::

            sage: P = Poset((divisors(15), attrcall("divides")), facade = False)

        Testing for wrapped elements::

            sage: all(P._vertex_to_element(P._element_to_vertex(x)) is x for x in P)
            True

        Testing for non-wrapped elements::

            sage: all(P._vertex_to_element(P._element_to_vertex(x)) is P(x) for x in divisors(15))
            True

        Testing for non-wrapped elements for a facade poset::

            sage: P = Poset((divisors(15), attrcall("divides")), facade = True)
            sage: all(P._vertex_to_element(P._element_to_vertex(x)) is x for x in P)
            True
        """
        if isinstance(element, self.element_class) and element.parent() is self:
            return element.vertex
        else:
            try:
                return self._element_to_vertex_dict[element]
            except KeyError:
                raise ValueError("element (=%s) not in poset" % element)

    def _vertex_to_element(self, vertex):
        """
        Return the element of ``self`` corresponding to the vertex
        ``vertex`` of the Hasse diagram.

        It is wrapped if ``self`` is not a facade poset.

        EXAMPLES::

            sage: P = Poset((divisors(15), attrcall("divides")), facade = False)
            sage: x = P._vertex_to_element(2)
            sage: x
            5
            sage: x.parent() is P
            True

            sage: P = Poset((divisors(15), attrcall("divides")), facade = True)
            sage: x = P._vertex_to_element(2)
            sage: x
            5
            sage: x.parent() is ZZ
            True
        """
        return self._list[vertex]

    def unwrap(self, element):
        """
        Return the element ``element`` of the poset ``self`` in
        unwrapped form.

        INPUT:

        - ``element`` -- an element of ``self``

        EXAMPLES::

            sage: P = Poset((divisors(15), attrcall("divides")), facade = False)
            sage: x = P.an_element(); x
            1
            sage: x.parent()
            Finite poset containing 4 elements
            sage: P.unwrap(x)
            1
            sage: P.unwrap(x).parent()
            Integer Ring

        For a non facade poset, this is equivalent to using the
        ``.element`` attribute::

            sage: P.unwrap(x) is x.element
            True

        For a facade poset, this does nothing::

            sage: P = Poset((divisors(15), attrcall("divides")), facade=True)
            sage: x = P.an_element()
            sage: P.unwrap(x) is x
            True

        This method is useful in code where we do not know if ``P`` is
        a facade poset or not.
        """
        if self._is_facade:
            return element
        else:
            return element.element

    def __contains__(self, x):
        r"""
        Return ``True`` if ``x`` is an element of the poset.

        TESTS::

            sage: from sage.combinat.posets.posets import FinitePoset
            sage: P5 = FinitePoset(DiGraph({(5,):[(4,1),(3,2)],
            ....:   (4,1):[(3,1,1),(2,2,1)],
            ....:   (3,2):[(3,1,1),(2,2,1)],
            ....:   (3,1,1):[(2,1,1,1)],
            ....:   (2,2,1):[(2,1,1,1)],
            ....:   (2,1,1,1):[(1,1,1,1,1)],
            ....:   (1,1,1,1,1):[]}))
            sage: x = P5.list()[3]
            sage: x in P5
            True

        For the sake of speed, an element with the right class and
        parent is assumed to be in this parent. This can possibly be
        counterfeited by feeding garbage to the constructor::

            sage: x = P5.element_class(P5, "a", 5)
            sage: x in P5
            True
        """
        if isinstance(x, self.element_class):
            return x.parent() is self
        return x in self._element_to_vertex_dict

    is_parent_of = __contains__

    def _element_constructor_(self, element):
        """
        Constructs an element of ``self``

        EXAMPLES::

            sage: from sage.combinat.posets.posets import FinitePoset
            sage: P = FinitePoset(DiGraph({0:[2,3], 1:[3,4], 2:[5], 3:[5], 4:[5]}), facade = False)
            sage: P(5)
            5
            sage: Q = FinitePoset(DiGraph({5:[2,3], 1:[3,4], 2:[0], 3:[0], 4:[0]}), facade = False)
            sage: Q(5)
            5

        Accessing the ``i``-th element of ``self`` as ``P[i]``::

            sage: P = FinitePoset(DiGraph({'a':['b','c'], 'b':['d'], 'c':['d'], 'd':[]}), facade = False)
            sage: P('a') == P[0]
            True
            sage: P('d') == P[-1]
            True

        TESTS::

            sage: P = Poset(DiGraph({0:[2,3], 1:[3,4], 2:[5], 3:[5], 4:[5]}), facade = False)
            sage: all(P(x) is x for x in P)
            True
            sage: P = Poset((divisors(15), attrcall("divides")), facade = True)
            sage: all(P(x) is x for x in P)
            True
        """
        try:
            return self._list[self._element_to_vertex_dict[element]]
        except KeyError:
            raise ValueError("%s is not an element of this poset"
                             % type(element))

    def __call__(self, element):
        """
        Creates elements of this poset

        This overrides the generic call method for all parents
        :meth:`Parent.__call__`, as a work around to allow for facade
        posets over plain Python objects (e.g. Python's
        int's). Indeed, the default __call__ method expects the result
        of :meth:`_element_constructor_` to be a Sage element (see
        :meth:`sage.structure.coerce_maps.DefaultConvertMap_unique._call_`)::

            sage: P = Poset(DiGraph({'d':['c','b'],'c':['a'],'b':['a']}),
            ....:           facade = True)
            sage: P('a')              # indirect doctest
            'a'
            sage: TestSuite(P).run()
            sage: P = Poset(((False, True), operator.eq), facade = True)
            sage: P(True)
            1
        """
        if self._is_facade and element in self._element_to_vertex_dict:
            return element
        return super(FinitePoset, self).__call__(element)

    def hasse_diagram(self):
        r"""
        Return the Hasse diagram of the poset as a Sage :class:`DiGraph`.

        The Hasse diagram is a directed graph where vertices are the
        elements of the poset and there is an edge from `u` to `v`
        whenever `v` covers `u` in the poset.

        If ``dot2tex`` is installed, then this sets the Hasse diagram's latex
        options to use the ``dot2tex`` formatting.

        EXAMPLES::

            sage: P = posets.DivisorLattice(12)
            sage: H = P.hasse_diagram(); H
            Digraph on 6 vertices
            sage: P.cover_relations()
            [[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
            sage: H.edges(labels=False)
            [(1, 2), (1, 3), (2, 4), (2, 6), (3, 6), (4, 12), (6, 12)]

        TESTS::

            sage: Poset().hasse_diagram()
            Digraph on 0 vertices

            sage: P = Poset((divisors(15), attrcall("divides")), facade=True)
            sage: H = P.hasse_diagram()
            sage: H.vertices()
            [1, 3, 5, 15]
            sage: H.edges()
            [(1, 3, None), (1, 5, None), (3, 15, None), (5, 15, None)]
            sage: H.set_latex_options(format="dot2tex")
            sage: view(H)  # optional - dot2tex, not tested (opens external window)
        """
        G = DiGraph(self._hasse_diagram).relabel(self._list, inplace=False)
        from sage.graphs.dot2tex_utils import have_dot2tex
        if have_dot2tex():
            G.set_latex_options(format='dot2tex',
                                prog='dot',
                                rankdir='up',)
        return G

    def _latex_(self):
        r"""
        Return a latex method for the poset.

        EXAMPLES::

            sage: P = Poset(([1,2], [[1,2]]), cover_relations = True)
            sage: print(P._latex_()) #optional - dot2tex graphviz
            \begin{tikzpicture}[>=latex,line join=bevel,]
            %%
            \node (node_...) at (6.0...bp,...bp) [draw,draw=none] {$...$};
              \node (node_...) at (6.0...bp,...bp) [draw,draw=none] {$...$};
              \draw [black,->] (node_...) ..controls (...bp,...bp) and (...bp,...bp)  .. (node_...);
            %
            \end{tikzpicture}
        """
        return self.hasse_diagram()._latex_()

    def _repr_(self):
        r"""
        Return a string representation of the poset.

        TESTS::

            sage: partitions_of_five = {(5,):[(4,1),(3,2)],
            ....:   (4,1):[(3,1,1),(2,2,1)],
            ....:   (3,2):[(3,1,1),(2,2,1)],
            ....:   (3,1,1):[(2,1,1,1)],
            ....:   (2,2,1):[(2,1,1,1)],
            ....:   (2,1,1,1):[(1,1,1,1,1)],
            ....:   (1,1,1,1,1):[]}
            sage: P5 = Poset(partitions_of_five)
            sage: P5._repr_()
            'Finite poset containing 7 elements'

            sage: M = MeetSemilattice([[1,2],[3],[3]])
            sage: M._repr_()
            'Finite meet-semilattice containing 3 elements'
        """
        s = "%s containing %s elements" % (self._desc, self._hasse_diagram.order())
        if self._with_linear_extension:
            s += " with distinguished linear extension"
        return s

    def _rich_repr_(self, display_manager, **kwds):
        """
        Rich Output Magic Method

        See :mod:`sage.repl.rich_output` for details.

        EXAMPLES::

            sage: from sage.repl.rich_output import get_display_manager
            sage: dm = get_display_manager()
            sage: Poset()._rich_repr_(dm, edge_labels=True)
            OutputPlainText container

        The ``supplemental_plot`` preference lets us control whether
        this object is shown as text or picture+text::

            sage: dm.preferences.supplemental_plot
            'never'
            sage: del dm.preferences.supplemental_plot
            sage: posets.ChainPoset(20)
            Finite lattice containing 20 elements (use the .plot() method to plot)
            sage: dm.preferences.supplemental_plot = 'never'
        """
        prefs = display_manager.preferences
        is_small = (0 < self.cardinality() < 20)
        can_plot = (prefs.supplemental_plot != 'never')
        plot_graph = can_plot and (prefs.supplemental_plot == 'always' or is_small)
        # Under certain circumstances we display the plot as graphics
        if plot_graph:
            plot_kwds = dict(kwds)
            plot_kwds.setdefault('title', repr(self))
            output = self.plot(**plot_kwds)._rich_repr_(display_manager)
            if output is not None:
                return output
        # create text for non-graphical output
        if can_plot:
            text = '{0} (use the .plot() method to plot)'.format(repr(self))
        else:
            text = repr(self)
        # latex() produces huge tikz environment, override
        tp = display_manager.types
        if (prefs.text == 'latex' and tp.OutputLatex in display_manager.supported_output()):
            return tp.OutputLatex(r'\text{{{0}}}'.format(text))
        return tp.OutputPlainText(text)

    def __iter__(self):
        """
        Iterates through the elements of a linear extension of the poset.

        EXAMPLES::

            sage: D = Poset({ 0:[1,2], 1:[3], 2:[3,4] })
            sage: sorted(D.__iter__())
            [0, 1, 2, 3, 4]
        """
        return iter(self._list)

    def sorted(self, l, allow_incomparable=True, remove_duplicates=False):
        """
        Return the list `l` sorted by the poset.

        INPUT:

        - ``l`` -- a list of elements of the poset
        - ``allow_incomparable`` -- a Boolean. If ``True`` (the default),
          return incomparable elements in some order; if ``False``, raise
          an error if ``l`` is not a chain of the poset.
        - ``remove_duplicates`` - a Boolean. If ``True``, remove duplicates
          from the output list.

        EXAMPLES::

            sage: P = posets.DivisorLattice(36)
            sage: P.sorted([1, 4, 1, 6, 2, 12])  # Random order for 4 and 6
            [1, 1, 2, 4, 6, 12]
            sage: P.sorted([1, 4, 1, 6, 2, 12], remove_duplicates=True)
            [1, 2, 4, 6, 12]
            sage: P.sorted([1, 4, 1, 6, 2, 12], allow_incomparable=False)
            Traceback (most recent call last):
            ...
            ValueError: the list contains incomparable elements

            sage: P = Poset({7:[1, 5], 1:[2, 6], 5:[3], 6:[3, 4]})
            sage: P.sorted([4, 1, 4, 5, 7])  # Random order for 1 and 5
            [7, 1, 5, 4, 4]
            sage: P.sorted([1, 4, 4, 7], remove_duplicates=True)
            [7, 1, 4]
            sage: P.sorted([4, 1, 4, 5, 7], allow_incomparable=False)
            Traceback (most recent call last):
            ...
            ValueError: the list contains incomparable elements

        TESTS::

            sage: P = posets.PentagonPoset()
            sage: P.sorted([], allow_incomparable=True, remove_duplicates=True)
            []
            sage: P.sorted([], allow_incomparable=False, remove_duplicates=True)
            []
            sage: P.sorted([], allow_incomparable=True, remove_duplicates=False)
            []
            sage: P.sorted([], allow_incomparable=False, remove_duplicates=False)
            []
        """
        v = [self._element_to_vertex(x) for x in l]
        if remove_duplicates:
            v = set(v)
        o = sorted(v)

        if not allow_incomparable:
            H = self._hasse_diagram
            if not all(H.is_lequal(a, b) for a, b in zip(o, o[1:])):
                raise ValueError("the list contains incomparable elements")

        return [self._vertex_to_element(x) for x in o]

    def linear_extension(self, linear_extension=None, check=True):
        r"""
        Return a linear extension of this poset.

        A linear extension of a finite poset `P` of size `n` is a total
        ordering `\pi := \pi_0 \pi_1 \ldots \pi_{n-1}` of its elements
        such that `i<j` whenever `\pi_i < \pi_j` in the poset `P`.

        INPUT:

        - ``linear_extension`` -- (default: ``None``) a list of the
          elements of ``self``
        - ``check`` -- a boolean (default: ``True``);
          whether to check that ``linear_extension`` is indeed a
          linear extension of ``self``.

        EXAMPLES::

            sage: P = Poset((divisors(15), attrcall("divides")), facade=True)

        Without optional argument, the default linear extension of the
        poset is returned, as a plain list::

            sage: P.linear_extension()
            [1, 3, 5, 15]

        Otherwise, a full-featured linear extension is constructed
        as an element of ``P.linear_extensions()``::

            sage: l = P.linear_extension([1,5,3,15]); l
            [1, 5, 3, 15]
            sage: type(l)
            <class 'sage.combinat.posets.linear_extensions.LinearExtensionsOfPoset_with_category.element_class'>
            sage: l.parent()
            The set of all linear extensions of Finite poset containing 4 elements

        By default, the linear extension is checked for correctness::

            sage: l = P.linear_extension([1,3,15,5])
            Traceback (most recent call last):
            ...
            ValueError: [1, 3, 15, 5] is not a linear extension of Finite poset containing 4 elements

        This can be disabled (at your own risks!) with::

            sage: P.linear_extension([1,3,15,5], check=False)
            [1, 3, 15, 5]

        .. SEEALSO:: :meth:`is_linear_extension`, :meth:`linear_extensions`

        .. TODO::

            - Is it acceptable to have those two features for a single method?

            - In particular, we miss a short idiom to get the default
              linear extension
        """
        L = self.linear_extensions()
        if linear_extension is not None:
            return L(linear_extension, check=check)
        return L(self._list, check=check)

    @cached_method
    def linear_extensions(self, facade=False):
        """
        Return the enumerated set of all the linear extensions of this poset.

        INPUT:

        - ``facade`` -- a boolean (default: ``False``);
          whether to return the linear extensions as plain lists

          .. warning::

            The ``facade`` option is not yet fully functional::

                sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
                sage: L = P.linear_extensions(facade=True); L
                The set of all linear extensions of Finite poset containing 6 elements with distinguished linear extension
                sage: L([1, 2, 3, 4, 6, 12])
                Traceback (most recent call last):
                ...
                TypeError: Cannot convert list to sage.structure.element.Element

        EXAMPLES::

            sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
            sage: P.list()
            [1, 2, 3, 4, 6, 12]
            sage: L = P.linear_extensions(); L
            The set of all linear extensions of Finite poset containing 6 elements with distinguished linear extension
            sage: l = L.an_element(); l
            [1, 2, 3, 4, 6, 12]
            sage: L.cardinality()
            5
            sage: L.list()
            [[1, 2, 3, 4, 6, 12],
             [1, 2, 4, 3, 6, 12],
             [1, 3, 2, 4, 6, 12],
             [1, 3, 2, 6, 4, 12],
             [1, 2, 3, 6, 4, 12]]

        Each element is aware that it is a linear extension of `P`::

            sage: type(l.parent())
            <class 'sage.combinat.posets.linear_extensions.LinearExtensionsOfPoset_with_category'>

        With ``facade=True``, the elements of ``L`` are plain lists instead::

            sage: L = P.linear_extensions(facade=True)
            sage: l = L.an_element()
            sage: type(l)
            <... 'list'>

        .. WARNING::

            In Sage <= 4.8, this function used to return a plain list
            of lists. To recover the previous functionality, please use::

                sage: L = list(P.linear_extensions(facade=True)); L
                [[1, 2, 3, 4, 6, 12],
                 [1, 2, 4, 3, 6, 12],
                 [1, 3, 2, 4, 6, 12],
                 [1, 3, 2, 6, 4, 12],
                 [1, 2, 3, 6, 4, 12]]
                sage: type(L[0])
                <... 'list'>

        .. SEEALSO:: :meth:`linear_extension`, :meth:`is_linear_extension`

        TESTS::

            sage: D = Poset({ 0:[1,2], 1:[3], 2:[3,4] })
            sage: list(D.linear_extensions())
            [[0, 1, 2, 3, 4], [0, 2, 1, 3, 4], [0, 2, 1, 4, 3], [0, 2, 4, 1, 3], [0, 1, 2, 4, 3]]
        """
        return self._lin_ext_type(self, facade=facade)

    def spectrum(self, a):
        r"""
        Return the `a`-spectrum of this poset.

        The `a`-spectrum in a poset `P` is the list of integers whose
        `i`-th position contains the number of linear extensions of `P`
        that have `a` in the `i`-th location.

        INPUT:

        - ``a`` -- an element of this poset

        OUTPUT:

        The `a`-spectrum of this poset, returned as a list.

        EXAMPLES::

            sage: P = posets.ChainPoset(5)
            sage: P.spectrum(2)
            [0, 0, 1, 0, 0]

            sage: P = posets.BooleanLattice(3)
            sage: P.spectrum(5)
            [0, 0, 0, 4, 12, 16, 16, 0]

            sage: P = posets.YoungDiagramPoset(Partition([3,2,1]))
            sage: P.spectrum((0,1))
            [0, 8, 6, 2, 0, 0]

            sage: P = posets.AntichainPoset(4)
            sage: P.spectrum(3)
            [6, 6, 6, 6]

        TESTS::

            sage: P = posets.ChainPoset(5)
            sage: P.spectrum(6)
            Traceback (most recent call last):
            ...
            ValueError: input element is not in poset
        """
        if a not in self:
            raise ValueError("input element is not in poset")

        a_spec = [0] * len(self)
        for L in self.linear_extensions():
            idx = L.index(a)
            a_spec[idx] += 1

        return a_spec

    def atkinson(self, a):
        r"""
        Return the `a`-spectrum of a poset whose Hasse diagram is
        cycle-free as an undirected graph.

        Given an element `a` in a poset `P`, the `a`-spectrum is the list of
        integers whose `i`-th term contains the number of linear extensions of
        `P` with element `a` located in the i-th position.

        INPUT:

        - ``self`` -- a poset whose Hasse diagram is a forest

        - ``a`` -- an element of the poset

        OUTPUT:

        The `a`-spectrum of this poset, returned as a list.

        EXAMPLES::

            sage: P = Poset({0: [2], 1: [2], 2: [3, 4], 3: [], 4: []})
            sage: P.atkinson(0)
            [2, 2, 0, 0, 0]

            sage: P = Poset({0: [1], 1: [2, 3], 2: [], 3: [], 4: [5, 6], 5: [], 6: []})
            sage: P.atkinson(5)
            [0, 10, 18, 24, 28, 30, 30]

            sage: P = posets.AntichainPoset(10)
            sage: P.atkinson(0)
            [362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880]

        TESTS::

            sage: P = posets.ChainPoset(5)
            sage: P.atkinson(6)
            Traceback (most recent call last):
            ...
            ValueError: input element is not in poset

            sage: P = posets.BooleanLattice(2)
            sage: P.atkinson(1)
            Traceback (most recent call last):
            ...
            ValueError: this poset is not a forest

        .. NOTE::

            This function is the implementation of the algorithm from [At1990]_.
        """
        if a not in self:
            raise ValueError("input element is not in poset")

        if not self._hasse_diagram.to_undirected().is_forest():
            raise ValueError("this poset is not a forest")

        n = self.cardinality()

        # Compute the component of this poset containing `a` and its complement
        components = self.connected_components()
        remainder_poset = Poset()

        for X in components:
            if a in X:
                main = X
            else:
                remainder_poset = remainder_poset.disjoint_union(X)

        hasse_a = main._element_to_vertex(a)
        a_spec = main._hasse_diagram._spectrum_of_tree(hasse_a)

        if not remainder_poset.cardinality():
            return a_spec

        b = remainder_poset.an_element()

        b_spec = remainder_poset.atkinson(b)
        n_lin_exts = sum(b_spec)

        new_a_spec = []
        k = main.cardinality()

        # Compute number of shuffles of linear extensions of the two posets
        for r in range(1, n + 1):
            new_a_spec.append(0)
            for i in range(max(1, r - n + k), min(r, k) + 1):
                k_val = binomial(r - 1, i - 1) * binomial(n - r, k - i)
                new_a_spec[-1] += k_val * a_spec[i - 1] * n_lin_exts
        return new_a_spec

    def is_linear_extension(self, l) -> bool:
        """
        Return whether ``l`` is a linear extension of ``self``.

        INPUT:

        - ``l`` -- a list (or iterable) containing all of the elements of ``self`` exactly once

        EXAMPLES::

            sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
            sage: P.list()
            [1, 2, 3, 4, 6, 12]
            sage: P.is_linear_extension([1, 2, 4, 3, 6, 12])
            True
            sage: P.is_linear_extension([1, 2, 4, 6, 3, 12])
            False

            sage: [p for p in Permutations(list(P)) if P.is_linear_extension(p)]
            [[1, 2, 3, 4, 6, 12],
             [1, 2, 3, 6, 4, 12],
             [1, 2, 4, 3, 6, 12],
             [1, 3, 2, 4, 6, 12],
             [1, 3, 2, 6, 4, 12]]
            sage: list(P.linear_extensions())
            [[1, 2, 3, 4, 6, 12],
             [1, 2, 4, 3, 6, 12],
             [1, 3, 2, 4, 6, 12],
             [1, 3, 2, 6, 4, 12],
             [1, 2, 3, 6, 4, 12]]

        .. NOTE::

            This is used and systematically tested in
            :class:`~sage.combinat.posets.linear_extensions.LinearExtensionsOfPosets`

        .. SEEALSO:: :meth:`linear_extension`, :meth:`linear_extensions`

        TESTS:

        Check that :trac:`15313` is fixed::

            sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
            sage: P.is_linear_extension([1,2,4,3,6,12,1337])
            False
            sage: P.is_linear_extension([1,2,4,3,6,666,12,1337])
            False
            sage: P = Poset(DiGraph(5))
            sage: P.is_linear_extension(['David', 'McNeil', 'La', 'Lamentable', 'Aventure', 'de', 'Simon', 'Wiesenthal'])
            False
        """
        index = {x: i for (i, x) in enumerate(l)}
        return (len(l) == self.cardinality() and
                all(x in index for x in self) and
                all(index[i] < index[j] for (i, j) in self.cover_relations()))

    def list(self):
        """
        List the elements of the poset. This just returns the result
        of :meth:`linear_extension`.

        EXAMPLES::

            sage: D = Poset({ 0:[1,2], 1:[3], 2:[3,4] }, facade = False)
            sage: D.list()
            [0, 1, 2, 3, 4]
            sage: type(D.list()[0])
            <class 'sage.combinat.posets.posets.FinitePoset_with_category.element_class'>
        """
        return list(self._list)

    def plot(self, label_elements=True, element_labels=None,
             layout='acyclic', cover_labels=None,
             **kwds):
        r"""
        Return a Graphic object for the Hasse diagram of the poset.

        If the poset is ranked, the plot uses the rank function for
        the heights of the elements.

        INPUT:

        - Options to change element look:

          * ``element_colors`` - a dictionary where keys are colors and values
            are lists of elements
          * ``element_color`` - a color for elements not set in
            ``element_colors``
          * ``element_shape`` - the shape of elements, like ``'s'`` for
            square; see https://matplotlib.org/api/markers_api.html for the list
          * ``element_size`` (default: 200) - the size of elements
          * ``label_elements`` (default: ``True``) - whether to display
            element labels
          * ``element_labels`` (default: ``None``) - a dictionary where keys
            are elements and values are labels to show

        - Options to change cover relation look:

          * ``cover_colors`` - a dictionary where keys are colors and values
            are lists of cover relations given as pairs of elements
          * ``cover_color`` - a color for elements not set in
            ``cover_colors``
          * ``cover_style`` - style for cover relations: ``'solid'``,
            ``'dashed'``, ``'dotted'`` or ``'dashdot'``
          * ``cover_labels`` - a dictionary, list or function representing
            labels of the covers of the poset. When set to ``None`` (default)
            no label is displayed on the edges of the Hasse Diagram.
          * ``cover_labels_background`` - a background color for cover
            relations. The default is "white". To achieve a transparent
            background use "transparent".

        - Options to change overall look:

          * ``figsize`` (default: 8) - size of the whole plot
          * ``title`` - a title for the plot
          * ``fontsize`` - fontsize for the title
          * ``border`` (default: ``False``) - whether to draw a border over the
            plot

        .. NOTE::

            All options of :meth:`GenericGraph.plot
            <sage.graphs.generic_graph.GenericGraph.plot>` are also available
            through this function.

        EXAMPLES:

        This function can be used without any parameters::

            sage: D12 = posets.DivisorLattice(12)
            sage: D12.plot()
            Graphics object consisting of 14 graphics primitives

        Just the abstract form of the poset; examples of relabeling::

            sage: D12.plot(label_elements=False)
            Graphics object consisting of 8 graphics primitives
            sage: d = {1: 0, 2: 'a', 3: 'b', 4: 'c', 6: 'd', 12: 1}
            sage: D12.plot(element_labels=d)
            Graphics object consisting of 14 graphics primitives
            sage: d = {i:str(factor(i)) for i in D12}
            sage: D12.plot(element_labels=d)
            Graphics object consisting of 14 graphics primitives

        Some settings for coverings::

            sage: d = {(a, b): b/a for a, b in D12.cover_relations()}
            sage: D12.plot(cover_labels=d, cover_color='gray', cover_style='dotted')
            Graphics object consisting of 21 graphics primitives

        To emphasize some elements and show some options::

            sage: L = LatticePoset({0: [1, 2, 3, 4], 1: [12], 2: [6, 7],
            ....:                   3: [5, 9], 4: [5, 6, 10, 11], 5: [13],
            ....:                   6: [12], 7: [12, 8, 9], 8: [13], 9: [13],
            ....:                   10: [12], 11: [12], 12: [13]})
            sage: F = L.frattini_sublattice()
            sage: F_internal = [c for c in F.cover_relations() if c in L.cover_relations()]
            sage: L.plot(figsize=12, border=True, element_shape='s',
            ....:        element_size=400, element_color='white',
            ....:        element_colors={'blue': F, 'green': L.double_irreducibles()},
            ....:        cover_color='lightgray', cover_colors={'black': F_internal},
            ....:        title='The Frattini\nsublattice in blue', fontsize=10)
            Graphics object consisting of 39 graphics primitives

        TESTS:

        We check that ``label_elements`` and ``element_labels`` are honored::

            sage: def get_plot_labels(P): return sorted(t.string for t in P if isinstance(t, sage.plot.text.Text))
            sage: P1 = Poset({ 0:[1,2], 1:[3], 2:[3,4] })
            sage: P2 = Poset({ 0:[1,2], 1:[3], 2:[3,4] }, facade=True)
            sage: get_plot_labels(P1.plot(label_elements=False))
            []
            sage: get_plot_labels(P1.plot(label_elements=True))
            [u'0', u'1', u'2', u'3', u'4']
            sage: element_labels = {0:'a', 1:'b', 2:'c', 3:'d', 4:'e'}
            sage: get_plot_labels(P1.plot(element_labels=element_labels))
            [u'a', u'b', u'c', u'd', u'e']
            sage: get_plot_labels(P2.plot(element_labels=element_labels))
            [u'a', u'b', u'c', u'd', u'e']

        The following checks that :trac:`18936` has been fixed and labels still work::

            sage: P = Poset({0: [1,2], 1:[3]})
            sage: heights = {1 : [0], 2 : [1], 3 : [2,3]}
            sage: P.plot(heights=heights)
            Graphics object consisting of 8 graphics primitives
            sage: elem_labels = {0 : 'a', 1 : 'b', 2 : 'c', 3 : 'd'}
            sage: P.plot(element_labels=elem_labels, heights=heights)
            Graphics object consisting of 8 graphics primitives

        The following checks that equal labels are allowed (:trac:`15206`)::

            sage: P = Poset({1: [2,3]})
            sage: labs = {i: P.rank(i) for i in range(1, 4)}
            sage: labs
            {1: 0, 2: 1, 3: 1}
            sage: P.plot(element_labels=labs)
            Graphics object consisting of 6 graphics primitives

        The following checks that non-hashable labels are allowed (:trac:`15206`)::

            sage: P = Poset({1: [2,3]})
            sage: labs = {1: [2, 3], 2: [], 3: []}
            sage: labs
            {1: [2, 3], 2: [], 3: []}
            sage: P.plot(element_labels=labs)
            Graphics object consisting of 6 graphics primitives

        Plot of the empty poset::

            sage: P = Poset({})
            sage: P.plot()
            Graphics object consisting of 0 graphics primitives
        """
        from collections import defaultdict
        graph = self.hasse_diagram()

        rename = {'element_color': 'vertex_color',
                  'element_colors': 'vertex_colors',
                  'element_size': 'vertex_size',
                  'element_shape': 'vertex_shape',
                  'cover_color': 'edge_color',
                  'cover_labels_background': 'edge_labels_background',
                  'cover_colors': 'edge_colors',
                  'cover_style': 'edge_style',
                  'border': 'graph_border'}
        for param in rename:
            tmp = kwds.pop(param, None)
            if tmp is not None:
                kwds[rename[param]] = tmp

        heights = kwds.pop('heights', None)
        if heights is None:
            rank_function = self.rank_function()
            if rank_function:  # use the rank function to set the heights
                heights = defaultdict(list)
                for i in self:
                    heights[rank_function(i)].append(i)
        # if relabeling is needed
        if label_elements and element_labels is not None:
            from sage.misc.element_with_label import ElementWithLabel
            relabeling = {self(element): ElementWithLabel(self(element), label)
                          for (element, label) in element_labels.items()}
            graph = graph.relabel(relabeling, inplace=False)
            if heights is not None:
                for key in heights:
                    heights[key] = [relabeling[i] for i in heights[key]]

        if cover_labels is not None:
            if callable(cover_labels):
                for (v, w) in graph.edges(labels=False):
                    graph.set_edge_label(v, w, cover_labels(v, w))
            elif isinstance(cover_labels, dict):
                for (v, w) in cover_labels:
                    graph.set_edge_label(self(v), self(w),
                                         cover_labels[(v, w)])
            else:
                for (v, w, l) in cover_labels:
                    graph.set_edge_label(self(v), self(w), l)
            cover_labels = True
        else:
            cover_labels = False

        return graph.plot(vertex_labels=label_elements,
                          edge_labels=cover_labels,
                          layout=layout,
                          heights=heights,
                          **kwds)

    def show(self, label_elements=True, element_labels=None,
             cover_labels=None, **kwds):
        """
        Displays the Hasse diagram of the poset.

        INPUT:

        - ``label_elements`` (default: ``True``) - whether to display
          element labels

        - ``element_labels`` (default: ``None``) - a dictionary of
          element labels

        - ``cover_labels`` - a dictionary, list or function representing labels
          of the covers of ``self``. When set to ``None`` (default) no label is
          displayed on the edges of the Hasse Diagram.

        .. NOTE::

            This method also accepts:

             - All options of :meth:`GenericGraph.plot
               <sage.graphs.generic_graph.GenericGraph.plot>`

             - All options of :meth:`Graphics.show
               <sage.plot.graphics.Graphics.show>`

        EXAMPLES::

            sage: D = Poset({ 0:[1,2], 1:[3], 2:[3,4] })
            sage: D.plot(label_elements=False)
            Graphics object consisting of 6 graphics primitives
            sage: D.show()
            sage: elm_labs = {0:'a', 1:'b', 2:'c', 3:'d', 4:'e'}
            sage: D.show(element_labels=elm_labs)

        One more example with cover labels::

            sage: P = posets.PentagonPoset()
            sage: P.show(cover_labels=lambda a, b: a - b)

        """
        # We split the arguments into those meant for plot() and those meant for show()
        #
        # The plot_kwds dictionary only contains the options that graphplot
        # understands. These options are removed from kwds at the same time.
        from sage.graphs.graph_plot import graphplot_options
        plot_kwds = {k: kwds.pop(k) for k in graphplot_options if k in kwds}

        self.plot(label_elements=label_elements,
                  element_labels=element_labels,
                  cover_labels=cover_labels,
                  **plot_kwds).show(**kwds)

    def level_sets(self):
        """
        Return elements grouped by maximal number of cover relations
        from a minimal element.

        This returns a list of lists ``l`` such that ``l[i]`` is the
        set of minimal elements of the poset obtained by removing the
        elements in ``l[0], l[1], ..., l[i-1]``. (In particular,
        ``l[0]`` is the set of minimal elements of ``self``.)

        Every level is an antichain of the poset.

        EXAMPLES::

            sage: P = Poset({0:[1,2],1:[3],2:[3],3:[]})
            sage: P.level_sets()
            [[0], [1, 2], [3]]

            sage: Q = Poset({0:[1,2], 1:[3], 2:[4], 3:[4]})
            sage: Q.level_sets()
            [[0], [1, 2], [3], [4]]

        .. SEEALSO::

            :meth:`dilworth_decomposition` to return elements grouped
            to chains.
        """
        return [[self._vertex_to_element(_) for _ in level] for level in
                self._hasse_diagram.level_sets()]

    def cover_relations(self):
        """
        Return the list of pairs ``[x, y]`` of elements of the poset such
        that ``y`` covers ``x``.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
            sage: P.cover_relations()
            [[1, 2], [0, 2], [2, 3], [3, 4]]
        """
        return [c for c in self.cover_relations_iterator()]

    @combinatorial_map(name="cover_relations_graph")
    def cover_relations_graph(self):
        """
        Return the (undirected) graph of cover relations.

        EXAMPLES::

            sage: P = Poset({0: [1, 2], 1: [3], 2: [3]})
            sage: G = P.cover_relations_graph(); G
            Graph on 4 vertices
            sage: G.has_edge(3, 1), G.has_edge(3, 0)
            (True, False)

        .. SEEALSO::

            :meth:`hasse_diagram`

        TESTS::

            sage: Poset().cover_relations_graph()
            Graph on 0 vertices

        Check that it is hashable and coincides with the Hasse diagram as a
        graph::

            sage: P = Poset({0: [1, 2], 1: [3], 2: [3]})
            sage: G = P.cover_relations_graph()
            sage: hash(G) == hash(G)
            True
            sage: G == Graph(P.hasse_diagram())
            True
        """
        from sage.graphs.graph import Graph
        return Graph(self.hasse_diagram(), immutable=True)

    def cover_relations_iterator(self):
        """
        Return an iterator over the cover relations of the poset.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
            sage: type(P.cover_relations_iterator())
            <... 'generator'>
            sage: [z for z in P.cover_relations_iterator()]
            [[1, 2], [0, 2], [2, 3], [3, 4]]
        """
        for u, v in self._hasse_diagram.edge_iterator(labels=False):
            yield [self._vertex_to_element(w) for w in (u, v)]

    def relations(self):
        r"""
        Return the list of all relations of the poset.

        A relation is a pair of elements `x` and `y` such that `x \leq y`
        in the poset.

        The number of relations is the dimension of the incidence
        algebra.

        OUTPUT:

        A list of pairs (each pair is a list), where the first element
        of the pair is less than or equal to the second element.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
            sage: P.relations()
            [[1, 1], [1, 2], [1, 3], [1, 4], [0, 0], [0, 2], [0, 3],
            [0, 4], [2, 2], [2, 3], [2, 4], [3, 3], [3, 4], [4, 4]]

        .. SEEALSO::

            :meth:`relations_number`, :meth:`relations_iterator`

        TESTS::

            sage: P = Poset()  # Test empty poset
            sage: P.relations()
            []

        AUTHOR:

        - Rob Beezer (2011-05-04)
        """
        return list(self.relations_iterator())

    def diamonds(self):
        r"""
        Return the list of diamonds of ``self``.

        A diamond is the following subgraph of the Hasse diagram::

                    z
                   / \
                  x   y
                   \ /
                    w

        Thus each edge represents a cover relation in the Hasse diagram.
        We represent this as the tuple `(w, x, y, z)`.

        OUTPUT:

        A tuple with

        - a list of all diamonds in the Hasse Diagram,
        - a boolean checking that every `w,x,y` that form a ``V``, there is a
          unique element `z`, which completes the diamond.

        EXAMPLES::

            sage: P = Poset({0: [1,2], 1: [3], 2: [3], 3: []})
            sage: P.diamonds()
            ([(0, 1, 2, 3)], True)

            sage: P = posets.YoungDiagramPoset(Partition([3, 2, 2]))
            sage: P.diamonds()
            ([((0, 0), (0, 1), (1, 0), (1, 1)), ((1, 0), (1, 1), (2, 0), (2, 1))], False)
        """
        diamonds, all_diamonds_completed = self._hasse_diagram.diamonds()
        return ([tuple(map(self._vertex_to_element, d)) for d in diamonds], all_diamonds_completed)

    def common_upper_covers(self, elmts):
        r"""
        Return all of the common upper covers of the elements ``elmts``.

        EXAMPLES::

            sage: P = Poset({0: [1,2], 1: [3], 2: [3], 3: []})
            sage: P.common_upper_covers([1, 2])
            [3]
        """
        vertices = list(map(self._element_to_vertex, elmts))
        return list(map(self._vertex_to_element, self._hasse_diagram.common_upper_covers(vertices)))

    def common_lower_covers(self, elmts):
        r"""
        Return all of the common lower covers of the elements ``elmts``.

        EXAMPLES::

            sage: P = Poset({0: [1,2], 1: [3], 2: [3], 3: []})
            sage: P.common_lower_covers([1, 2])
            [0]
        """
        vertices = list(map(self._element_to_vertex, elmts))
        return list(map(self._vertex_to_element, self._hasse_diagram.common_lower_covers(vertices)))

    def meet(self, x, y):
        r"""
        Return the meet of two elements ``x, y`` in the poset if the meet
        exists; and ``None`` otherwise.

        EXAMPLES::

            sage: D = Poset({1:[2,3], 2:[4], 3:[4]})
            sage: D.meet(2, 3)
            1
            sage: P = Poset({'a':['b', 'c'], 'b':['e', 'f'], 'c':['f', 'g'],
            ....:            'd':['f', 'g']})
            sage: P.meet('a', 'b')
            'a'
            sage: P.meet('e', 'a')
            'a'
            sage: P.meet('c', 'b')
            'a'
            sage: P.meet('e', 'f')
            'b'
            sage: P.meet('e', 'g')
            'a'
            sage: P.meet('c', 'd') is None
            True
            sage: P.meet('g', 'f') is None
            True
        """
        i, j = map(self._element_to_vertex, (x, y))
        mt = self._hasse_diagram._meet
        if mt[i, j] == -1:
            return None
        else:
            return self._vertex_to_element(mt[i, j])

    def join(self, x, y):
        r"""
        Return the join of two elements ``x, y`` in the poset if the join
        exists; and ``None`` otherwise.

        EXAMPLES::

            sage: D = Poset({1:[2,3], 2:[4], 3:[4]})
            sage: D.join(2, 3)
            4
            sage: P = Poset({'e':['b'], 'f':['b', 'c', 'd'], 'g':['c', 'd'],
            ....:            'b':['a'], 'c':['a']})
            sage: P.join('a', 'b')
            'a'
            sage: P.join('e', 'a')
            'a'
            sage: P.join('c', 'b')
            'a'
            sage: P.join('e', 'f')
            'b'
            sage: P.join('e', 'g')
            'a'
            sage: P.join('c', 'd') is None
            True
            sage: P.join('g', 'f') is None
            True
        """
        i, j = map(self._element_to_vertex, (x, y))
        jn = self._hasse_diagram._join
        if jn[i,j] == -1:
            return None
        else:
            return self._vertex_to_element(jn[i,j])

    def is_d_complete(self) -> bool:
        r"""
        Return ``True`` if a poset is d-complete and ``False`` otherwise.

        .. SEEALSO::

            - :mod:`~sage.combinat.posets.d_complete`

        EXAMPLES::

            sage: from sage.combinat.posets.posets import FinitePoset
            sage: A = Poset({0: [1,2]})
            sage: A.is_d_complete()
            False

            sage: from sage.combinat.posets.poset_examples import Posets
            sage: B = Posets.DoubleTailedDiamond(3)
            sage: B.is_d_complete()
            True

            sage: C = Poset({0: [2], 1: [2], 2: [3, 4], 3: [5], 4: [5], 5: [6]})
            sage: C.is_d_complete()
            False

            sage: D = Poset({0: [1, 2], 1: [4], 2: [4], 3: [4]})
            sage: D.is_d_complete()
            False

            sage: P = Posets.YoungDiagramPoset(Partition([3, 2, 2]), dual=True)
            sage: P.is_d_complete()
            True
        """
        min_diamond = {}  # Maps max of double-tailed diamond to min of double-tailed diamond
        max_diamond = {}  # Maps min of double-tailed diamond to max of double-tailed diamond

        H = self._hasse_diagram

        diamonds, all_diamonds_completed = H.diamonds()  # Tuples of four elements that are diamonds

        if not all_diamonds_completed:
            return False

        diamond_index = {}  # Map max elmt of double tailed diamond to index of diamond

        # Find all the double-tailed diamonds and map the mins and maxes
        for index, d in enumerate(diamonds):
            min_diamond[d[3]] = d[0]
            max_diamond[d[0]] = d[3]
            diamond_index[d[3]] = index

            min_elmt = d[0]
            max_elmt = d[3]

            if len(H.neighbors_in(max_elmt)) != 2:
                # Top of a diamond cannot cover anything but the two side elements
                return False

            while True:
                potential_min = H.neighbors_in(min_elmt)
                potential_max = H.neighbors_out(max_elmt)
                max_dk_minus = max_elmt

                # Check if any of these make a longer double tailed diamond
                found_diamond = False
                for mn in potential_min:
                    if len(H.all_paths(mn, max_dk_minus)) > 2:
                        continue
                    for mx in potential_max:
                        if len(H.all_paths(mn, mx)) == 2:
                            if len(H.neighbors_in(mx)) != 1:
                                # Max element covers something outside of double tailed diamond
                                return False
                            # Success
                            if mx in min_diamond or mn in max_diamond:
                                # Two double tail diamonds that differ by minimal element
                                # or the dual
                                return False
                            min_elmt = mn
                            max_elmt = mx

                            min_diamond[mx] = mn
                            max_diamond[mn] = mx
                            diamond_index[mx] = index
                            found_diamond = True
                if not found_diamond:
                    break

        return True

    def slant_sum(self, p, element, p_element):
        r"""
        Return the slant sum poset of posets ``self`` and ``p`` by
        connecting them with a cover relation ``(p_element, element)``.

        .. NOTE::

            The element names of ``self`` and ``p`` must be distinct.

        INPUT:

        - ``p`` -- the poset used for the slant sum
        - ``element`` -- the element of ``self`` that is the top of
          the new cover relation
        - ``p_element`` -- the element of ``p`` that is the bottom of
          the new cover relation

        EXAMPLES::

            sage: R = posets.RibbonPoset(5, [1,2])
            sage: H = Poset([[5, 6, 7], [(5, 6), (6,7)]])
            sage: SS = R.slant_sum(H, 3, 7)
            sage: all(cr in SS.cover_relations() for cr in R.cover_relations())
            True
            sage: all(cr in SS.cover_relations() for cr in H.cover_relations())
            True
            sage: SS.covers(7, 3)
            True
        """
        elements = p._elements + self._elements
        cover_relations = p.cover_relations() + self.cover_relations()
        cover_relations.append((p_element, element))
        return Poset([elements, cover_relations])

    def intervals_poset(self):
        r"""
        Return the natural partial order on the set of intervals of the poset.

        OUTPUT:

        a finite poset

        The poset of intervals of a poset `P` has the set of intervals `[x,y]`
        in `P` as elements, endowed with the order relation defined by
        `[x_1,y_1] \leq [x_2,y_2]` if and only if `x_1 \leq x_2` and
        `y_1 \leq y_2`.

        This is also called `P` to the power *2*, meaning
        the poset of poset-morphisms from the 2-chain to `P`.

        If `P` is a lattice, the result is also a lattice.

        EXAMPLES::

            sage: P = Poset({0:[1]})
            sage: P.intervals_poset()
            Finite poset containing 3 elements

            sage: P = posets.PentagonPoset()
            sage: P.intervals_poset()
            Finite lattice containing 13 elements

        TESTS::

            sage: P = Poset({})
            sage: P.intervals_poset()
            Finite poset containing 0 elements

            sage: P = Poset({0:[]})
            sage: P.intervals_poset()
            Finite poset containing 1 elements

            sage: P = Poset({0:[], 1:[]})
            sage: P.intervals_poset().is_isomorphic(P)
            True
        """
        from sage.combinat.posets.lattices import (LatticePoset,
                                                   FiniteLatticePoset)
        if isinstance(self, FiniteLatticePoset):
            constructor = LatticePoset
        else:
            constructor = Poset

        ints = [tuple(u) for u in self.relations()]

        covers = []
        for (a, b) in ints:
            covers.extend([[(a, b), (a, bb)] for bb in self.upper_covers(b)])
            if a != b:
                covers.extend([[(a, b), (aa, b)] for aa in self.upper_covers(a)
                               if self.le(aa, b)])

        dg = DiGraph([ints, covers], format="vertices_and_edges")
        return constructor(dg, cover_relations=True)

    def relations_iterator(self, strict=False):
        r"""
        Return an iterator for all the relations of the poset.

        A relation is a pair of elements `x` and `y` such that `x \leq y`
        in the poset.

        INPUT:

        - ``strict`` -- a boolean (default ``False``) if ``True``, returns
          an iterator over relations `x < y`, excluding all
          relations `x \leq x`.

        OUTPUT:

        A generator that produces pairs (each pair is a list), where the
        first element of the pair is less than or equal to the second element.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
            sage: it = P.relations_iterator()
            sage: type(it)
            <... 'generator'>
            sage: next(it), next(it)
            ([1, 1], [1, 2])

            sage: P = posets.PentagonPoset()
            sage: list(P.relations_iterator(strict=True))
            [[0, 1], [0, 2], [0, 4], [0, 3], [1, 4], [2, 3], [2, 4], [3, 4]]

        .. SEEALSO::

            :meth:`relations_number`, :meth:`relations`.

        AUTHOR:

        - Rob Beezer (2011-05-04)
        """
        elements = self._elements
        hd = self._hasse_diagram
        if strict:
            for i in hd:
                for j in hd.breadth_first_search(i):
                    if i != j:
                        yield [elements[i], elements[j]]
        else:
            for i in hd:
                for j in hd.breadth_first_search(i):
                    yield [elements[i], elements[j]]

    def relations_number(self):
        r"""
        Return the number of relations in the poset.

        A relation is a pair of elements `x` and `y` such that `x\leq y`
        in the poset.

        Relations are also often called intervals. The number of
        intervals is the dimension of the incidence algebra.

        EXAMPLES::

            sage: P = posets.PentagonPoset()
            sage: P.relations_number()
            13

            sage: posets.TamariLattice(4).relations_number()
            68

        .. SEEALSO::

            :meth:`relations_iterator`, :meth:`relations`

        TESTS::

            sage: Poset().relations_number()
            0
        """
        return sum(1 for _ in self.relations_iterator())

    # Maybe this should also be deprecated.
    intervals_number = relations_number

    def is_incomparable_chain_free(self, m, n=None) -> bool:
        r"""
        Return ``True`` if the poset is `(m+n)`-free, and ``False`` otherwise.

        A poset is `(m+n)`-free if there is no incomparable chains of
        lengths `m` and `n`. Three cases have special name
        (see [EnumComb1]_, exercise 3.15):

        - ''interval order'' is `(2+2)`-free
        - ''semiorder'' (or ''unit interval order'') is `(1+3)`-free and
          `(2+2)`-free
        - ''weak order'' is `(1+2)`-free.

        INPUT:

        - ``m``, ``n`` - positive integers

        It is also possible to give a list of integer pairs as argument.
        See below for an example.

        EXAMPLES::

            sage: B3 = posets.BooleanLattice(3)
            sage: B3.is_incomparable_chain_free(1, 3)
            True
            sage: B3.is_incomparable_chain_free(2, 2)
            False

            sage: IP6 = posets.IntegerPartitions(6)
            sage: IP6.is_incomparable_chain_free(1, 3)
            False
            sage: IP6.is_incomparable_chain_free(2, 2)
            True

        A list of pairs as an argument::

            sage: B3.is_incomparable_chain_free([[1, 3], [2, 2]])
            False

        We show how to get an incomparable chain pair::

            sage: P = posets.PentagonPoset()
            sage: chains_1_2 = Poset({0:[], 1:[2]})
            sage: incomps = P.isomorphic_subposets(chains_1_2)[0]
            sage: sorted(incomps.list()), incomps.cover_relations()
            ([1, 2, 3], [[2, 3]])

        TESTS::

            sage: Poset().is_incomparable_chain_free(1,1)  # Test empty poset
            True

            sage: [len([p for p in Posets(n) if p.is_incomparable_chain_free(((3, 1), (2, 2)))]) for n in range(6)] # long time
            [1, 1, 2, 5, 14, 42]

            sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
            sage: Q.is_incomparable_chain_free(2, 20/10)
            True
            sage: Q.is_incomparable_chain_free(2, pi)
            Traceback (most recent call last):
            ...
            TypeError: 2 and pi must be integers.
            sage: Q.is_incomparable_chain_free(2, -1)
            Traceback (most recent call last):
            ...
            ValueError: 2 and -1 must be positive integers.
            sage: P = Poset(((0, 1, 2, 3, 4), ((0, 1), (1, 2), (0, 3), (4, 2))))
            sage: P.is_incomparable_chain_free((3, 1))
            Traceback (most recent call last):
            ...
            TypeError: (3, 1) is not a tuple of tuples.
            sage: P.is_incomparable_chain_free([3, 1], [2, 2])
            Traceback (most recent call last):
            ...
            TypeError: [3, 1] and [2, 2] must be integers.
            sage: P.is_incomparable_chain_free([[3, 1], [2, 2]])
            True
            sage: P.is_incomparable_chain_free(([3, 1], [2, 2]))
            True
            sage: P.is_incomparable_chain_free([3, 1], 2)
            Traceback (most recent call last):
            ...
            TypeError: [3, 1] and 2 must be integers.
            sage: P.is_incomparable_chain_free(([3, 1], [2, 2, 2]))
            Traceback (most recent call last):
            ...
            ValueError: '([3, 1], [2, 2, 2])' is not a tuple of length-2 tuples.

        AUTHOR:

        - Eric Rowland (2013-05-28)
        """
        if n is None:
            try:
                chain_pairs = [tuple(chain_pair) for chain_pair in m]
            except TypeError:
                raise TypeError("%s is not a tuple of tuples." % str(tuple(m)))
            if not all(len(chain_pair) == 2 for chain_pair in chain_pairs):
                raise ValueError("%r is not a tuple of length-2 tuples." % str(tuple(m)))
            chain_pairs = sorted(chain_pairs, key=min)
        else:
            chain_pairs = [(m, n)]

        if chain_pairs:
            closure = self._hasse_diagram.transitive_closure()
        for m, n in chain_pairs:
            try:
                m, n = Integer(m), Integer(n)
            except TypeError:
                raise TypeError("%s and %s must be integers." % (m, n))
            if m < 1 or n < 1:
                raise ValueError("%s and %s must be positive integers." % (m, n))
            twochains = digraphs.TransitiveTournament(m) + digraphs.TransitiveTournament(n)
            if closure.subgraph_search(twochains, induced=True) is not None:
                return False
        return True

    def is_lequal(self, x, y) -> bool:
        """
        Return ``True`` if `x` is less than or equal to `y` in the poset, and
        ``False`` otherwise.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
            sage: P.is_lequal(2, 4)
            True
            sage: P.is_lequal(2, 2)
            True
            sage: P.is_lequal(0, 1)
            False
            sage: P.is_lequal(3, 2)
            False

        .. SEEALSO:: :meth:`is_less_than`, :meth:`is_gequal`.
        """
        i = self._element_to_vertex(x)
        j = self._element_to_vertex(y)
        return (self._hasse_diagram.is_lequal(i, j))

    le = is_lequal

    def is_less_than(self, x, y) -> bool:
        """
        Return ``True`` if `x` is less than but not equal to `y` in the poset,
        and ``False`` otherwise.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
            sage: P.is_less_than(1, 3)
            True
            sage: P.is_less_than(0, 1)
            False
            sage: P.is_less_than(2, 2)
            False

        For non-facade posets also ``<`` works::

            sage: P = Poset({3: [1, 2]}, facade=False)
            sage: P(1) < P(2)
            False

        .. SEEALSO:: :meth:`is_lequal`, :meth:`is_greater_than`.
        """
        i = self._element_to_vertex(x)
        j = self._element_to_vertex(y)
        return self._hasse_diagram.is_less_than(i, j)

    lt = is_less_than

    def is_gequal(self, x, y) -> bool:
        """
        Return ``True`` if `x` is greater than or equal to `y` in the poset,
        and ``False`` otherwise.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
            sage: P.is_gequal(3, 1)
            True
            sage: P.is_gequal(2, 2)
            True
            sage: P.is_gequal(0, 1)
            False

        .. SEEALSO:: :meth:`is_greater_than`, :meth:`is_lequal`.
        """
        i = self._element_to_vertex(x)
        j = self._element_to_vertex(y)
        return (self._hasse_diagram.is_lequal(j, i))

    ge = is_gequal

    def is_greater_than(self, x, y) -> bool:
        """
        Return ``True`` if `x` is greater than but not equal to `y` in the
        poset, and ``False`` otherwise.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
            sage: P.is_greater_than(3, 1)
            True
            sage: P.is_greater_than(1, 2)
            False
            sage: P.is_greater_than(3, 3)
            False
            sage: P.is_greater_than(0, 1)
            False

        For non-facade posets also ``>`` works::

            sage: P = Poset({3: [1, 2]}, facade=False)
            sage: P(2) > P(3)
            True

        .. SEEALSO:: :meth:`is_gequal`, :meth:`is_less_than`.
        """
        i = self._element_to_vertex(x)
        j = self._element_to_vertex(y)
        return self._hasse_diagram.is_less_than(j, i)

    gt = is_greater_than

    def compare_elements(self, x, y):
        r"""
        Compare `x` and `y` in the poset.

        - If `x < y`, return ``-1``.
        - If `x = y`, return ``0``.
        - If `x > y`, return ``1``.
        - If `x` and `y` are not comparable, return ``None``.

        EXAMPLES::

            sage: P = Poset([[1, 2], [4], [3], [4], []])
            sage: P.compare_elements(0, 0)
            0
            sage: P.compare_elements(0, 4)
            -1
            sage: P.compare_elements(4, 0)
            1
            sage: P.compare_elements(1, 2) is None
            True
        """
        i, j = map(self._element_to_vertex, (x, y))
        if i == j:
            return 0
        elif self._hasse_diagram.is_less_than(i, j):
            return -1
        elif self._hasse_diagram.is_less_than(j, i):
            return 1
        else:
            return None

    def minimal_elements(self):
        """
        Return the list of the minimal elements of the poset.

        EXAMPLES::

            sage: P = Poset({0:[3],1:[3],2:[3],3:[4],4:[]})
            sage: P(0) in P.minimal_elements()
            True
            sage: P(1) in P.minimal_elements()
            True
            sage: P(2) in P.minimal_elements()
            True

        .. SEEALSO:: :meth:`maximal_elements`.
        """
        return [self._vertex_to_element(_) for _ in self._hasse_diagram.minimal_elements()]

    def maximal_elements(self):
        """
        Return the list of the maximal elements of the poset.

        EXAMPLES::

            sage: P = Poset({0:[3],1:[3],2:[3],3:[4],4:[]})
            sage: P.maximal_elements()
            [4]

        .. SEEALSO:: :meth:`minimal_elements`.
        """
        return [self._vertex_to_element(_) for _ in self._hasse_diagram.maximal_elements()]

    def bottom(self):
        """
        Return the unique minimal element of the poset, if it exists.

        EXAMPLES::

            sage: P = Poset({0:[3],1:[3],2:[3],3:[4],4:[]})
            sage: P.bottom() is None
            True
            sage: Q = Poset({0:[1],1:[]})
            sage: Q.bottom()
            0

        .. SEEALSO:: :meth:`has_bottom`, :meth:`top`
        """
        hasse_bot = self._hasse_diagram.bottom()
        if hasse_bot is None:
            return None
        else:
            return self._vertex_to_element(hasse_bot)

    def has_bottom(self):
        """
        Return ``True`` if the poset has a unique minimal element, and
        ``False`` otherwise.

        EXAMPLES::

            sage: P = Poset({0:[3], 1:[3], 2:[3], 3:[4], 4:[]})
            sage: P.has_bottom()
            False
            sage: Q = Poset({0:[1], 1:[]})
            sage: Q.has_bottom()
            True

        .. SEEALSO::

            - Dual Property: :meth:`has_top`
            - Stronger properties: :meth:`is_bounded`
            - Other: :meth:`bottom`

        TESTS::

            sage: Poset().has_top()  # Test empty poset
            False
        """
        return self._hasse_diagram.has_bottom()

    def top(self):
        """
        Return the unique maximal element of the poset, if it exists.

        EXAMPLES::

            sage: P = Poset({0:[3],1:[3],2:[3],3:[4,5],4:[],5:[]})
            sage: P.top() is None
            True
            sage: Q = Poset({0:[1],1:[]})
            sage: Q.top()
            1

        .. SEEALSO:: :meth:`has_top`, :meth:`bottom`

        TESTS::

            sage: R = Poset([[0],[]])
            sage: R.list()
            [0]
            sage: R.top() #Trac #10776
            0
        """
        hasse_top = self._hasse_diagram.top()
        if hasse_top is None:
            return None
        else:
            return self._vertex_to_element(hasse_top)

    def has_top(self):
        """
        Return ``True`` if the poset has a unique maximal element, and
        ``False`` otherwise.

        EXAMPLES::

            sage: P = Poset({0:[3], 1:[3], 2:[3], 3:[4, 5], 4:[], 5:[]})
            sage: P.has_top()
            False
            sage: Q = Poset({0:[3], 1:[3], 2:[3], 3:[4], 4:[]})
            sage: Q.has_top()
            True

        .. SEEALSO::

            - Dual Property: :meth:`has_bottom`
            - Stronger properties: :meth:`is_bounded`
            - Other: :meth:`top`

        TESTS::

            sage: Poset().has_top()  # Test empty poset
            False
        """
        return self._hasse_diagram.has_top()

    def height(self, certificate=False):
        """
        Return the height (number of elements in a longest chain) of the poset.

        INPUT:

        - ``certificate`` -- (default: ``False``) whether to return
          a certificate

        OUTPUT:

        - If ``certificate=True`` return ``(h, c)``, where ``h`` is the
          height and ``c`` is a chain of maximum cardinality.
          If ``certificate=False`` return only the height.

        EXAMPLES::

            sage: P = Poset({0: [1], 2: [3, 4], 4: [5, 6]})
            sage: P.height()
            3
            sage: posets.PentagonPoset().height(certificate=True)
            (4, [0, 2, 3, 4])

        TESTS::

            sage: Poset().height()
            0
        """
        if not certificate:
            return self.rank() + 1

        levels = self.level_sets()
        height = len(levels)
        if height == 0:
            return (0, [])
        n = height - 2
        previous = levels[-1][0]
        max_chain = [previous]

        while n >= 0:
            for i in levels[n]:
                if self.covers(i, previous):
                    break
            max_chain.append(i)
            previous = i
            n -= 1

        max_chain.reverse()
        return (height, max_chain)

    def has_isomorphic_subposet(self, other):
        """
        Return ``True`` if the poset contains a subposet isomorphic to
        ``other``.

        By subposet we mean that there exist a set ``X`` of elements such
        that ``self.subposet(X)`` is isomorphic to ``other``.

        INPUT:

        - ``other`` -- a finite poset

        EXAMPLES::

            sage: D = Poset({1:[2,3], 2:[4], 3:[4]})
            sage: T = Poset({1:[2,3], 2:[4,5], 3:[6,7]})
            sage: N5 = posets.PentagonPoset()

            sage: N5.has_isomorphic_subposet(T)
            False
            sage: N5.has_isomorphic_subposet(D)
            True

            sage: len([P for P in Posets(5) if P.has_isomorphic_subposet(D)])
            11

        """
        if not hasattr(other, 'hasse_diagram'):
            raise TypeError("'other' is not a finite poset")
        if self._hasse_diagram.transitive_closure().subgraph_search(other._hasse_diagram.transitive_closure(), induced=True) is None:
            return False
        return True

    def is_bounded(self) -> bool:
        """
        Return ``True`` if the poset is bounded, and ``False`` otherwise.

        A poset is bounded if it contains both a unique maximal element
        and a unique minimal element.

        EXAMPLES::

            sage: P = Poset({0:[3], 1:[3], 2:[3], 3:[4, 5], 4:[], 5:[]})
            sage: P.is_bounded()
            False
            sage: Q = posets.DiamondPoset(5)
            sage: Q.is_bounded()
            True

        .. SEEALSO::

            - Weaker properties: :meth:`has_bottom`, :meth:`has_top`
            - Other: :meth:`with_bounds`, :meth:`without_bounds`

        TESTS::

            sage: Poset().is_bounded()  # Test empty poset
            False
            sage: Poset({1:[]}).is_bounded()  # Here top == bottom
            True
            sage: ( len([P for P in Posets(5) if P.is_bounded()]) ==
            ....: Posets(3).cardinality() )
            True
        """
        return self._hasse_diagram.is_bounded()

    def is_chain(self) -> bool:
        """
        Return ``True`` if the poset is totally ordered ("chain"), and
        ``False`` otherwise.

        EXAMPLES::

            sage: I = Poset({0:[1], 1:[2], 2:[3], 3:[4]})
            sage: I.is_chain()
            True

            sage: II = Poset({0:[1], 2:[3]})
            sage: II.is_chain()
            False

            sage: V = Poset({0:[1, 2]})
            sage: V.is_chain()
            False

        TESTS::

            sage: [len([P for P in Posets(n) if P.is_chain()]) for n in range(5)]
            [1, 1, 1, 1, 1]
        """
        return self._hasse_diagram.is_chain()

    def is_chain_of_poset(self, elms, ordered=False) -> bool:
        """
        Return ``True`` if ``elms`` is a chain of the poset,
        and ``False`` otherwise.

        Set of elements are a *chain* of a poset if they are comparable
        to each other.

        INPUT:

        - ``elms`` -- a list or other iterable containing some elements
          of the poset

        - ``ordered`` -- a Boolean. If ``True``, then return ``True``
          only if elements in ``elms`` are strictly increasing in the
          poset; this makes no sense if ``elms`` is a set. If ``False``
          (the default), then elements can be repeated and be in any
          order.

        EXAMPLES::

            sage: P = Poset((divisors(12), attrcall("divides")))
            sage: sorted(P.list())
            [1, 2, 3, 4, 6, 12]
            sage: P.is_chain_of_poset([12, 3])
            True
            sage: P.is_chain_of_poset({3, 4, 12})
            False
            sage: P.is_chain_of_poset([12, 3], ordered=True)
            False
            sage: P.is_chain_of_poset((1, 1, 3))
            True
            sage: P.is_chain_of_poset((1, 1, 3), ordered=True)
            False
            sage: P.is_chain_of_poset((1, 3), ordered=True)
            True

        TESTS::

            sage: P = posets.BooleanLattice(4)
            sage: P.is_chain_of_poset([])
            True
            sage: P.is_chain_of_poset((1,3,7,15,14))
            False
            sage: P.is_chain_of_poset({10})
            True
            sage: P.is_chain_of_poset([32])
            Traceback (most recent call last):
            ...
            ValueError: element (=32) not in poset
        """
        if ordered:
            sorted_o = elms
            return all(self.lt(a, b) for a, b in zip(sorted_o, sorted_o[1:]))
        else:
            # _element_to_vertex can be assumed to be a linear extension
            # of the poset according to the documentation of class
            # HasseDiagram.
            sorted_o = sorted(elms, key=self._element_to_vertex)
            return all(self.le(a, b) for a, b in zip(sorted_o, sorted_o[1:]))

    def is_antichain_of_poset(self, elms):
        """
        Return ``True`` if ``elms`` is an antichain of the poset
        and ``False`` otherwise.

        Set of elements are an *antichain* of a poset if they are
        pairwise incomparable.

        EXAMPLES::

            sage: P = posets.BooleanLattice(5)
            sage: P.is_antichain_of_poset([3, 5, 7])
            False
            sage: P.is_antichain_of_poset([3, 5, 14])
            True

        TESTS::

            sage: P = posets.PentagonPoset()
            sage: P.is_antichain_of_poset([])
            True
            sage: P.is_antichain_of_poset([0])
            True
            sage: P.is_antichain_of_poset([1, 2, 1])
            True

        Check :trac:`19078`::

            sage: P.is_antichain_of_poset([0, 1, 'junk'])
            Traceback (most recent call last):
            ...
            ValueError: element (=junk) not in poset
        """
        elms_H = [self._element_to_vertex(e) for e in elms]
        return self._hasse_diagram.is_antichain_of_poset(elms_H)

    def is_connected(self) -> bool:
        """
        Return ``True`` if the poset is connected, and ``False`` otherwise.

        A poset is connected if its Hasse diagram is connected.

        If a poset is not connected, then it can be divided to parts
        `S_1` and `S_2` so that every element of `S_1` is incomparable to
        every element of `S_2`.

        EXAMPLES::

            sage: P = Poset({1:[2, 3], 3:[4, 5]})
            sage: P.is_connected()
            True

            sage: P = Poset({1:[2, 3], 3:[4, 5], 6:[7, 8]})
            sage: P.is_connected()
            False

        .. SEEALSO:: :meth:`connected_components`

        TESTS::

            sage: Poset().is_connected()  # Test empty poset
            True
        """
        return self._hasse_diagram.is_connected()

    def is_series_parallel(self) -> bool:
        """
        Return ``True`` if the poset is series-parallel, and ``False``
        otherwise.

        A poset is *series-parallel* if it can be built up from one-element
        posets using the operations of disjoint union and ordinal
        sum. This is also called *N-free* property: every poset that is not
        series-parallel contains a subposet isomorphic to the 4-element
        N-shaped poset where `a > c, d` and `b > d`.

        .. NOTE::

            Some papers use the term N-free for posets having no
            N-shaped poset as a *cover-preserving subposet*. This definition
            is not used here.

        See :wikipedia:`Series-parallel partial order`.

        EXAMPLES::

            sage: VA = Poset({1: [2, 3], 4: [5], 6: [5]})
            sage: VA.is_series_parallel()
            True
            sage: big_N = Poset({1: [2, 4], 2: [3], 4:[7], 5:[6], 6:[7]})
            sage: big_N.is_series_parallel()
            False

        TESTS::

            sage: Poset().is_series_parallel()
            True
        """
        if self.cardinality() < 4:
            return True
        if not self.is_connected():
            return all(part.is_series_parallel() for part in
                       self.connected_components())
        parts = self.ordinal_summands()
        if len(parts) == 1:
            return False
        return all(part.is_series_parallel() for part in parts)

    def is_EL_labelling(self, f, return_raising_chains=False):
        r"""
        Return ``True`` if ``f`` is an EL labelling of ``self``.

        A labelling `f` of the edges of the Hasse diagram of a poset
        is called an EL labelling (edge lexicographic labelling) if
        for any two elements `u` and `v` with `u \leq v`,

            - there is a unique `f`-raising chain from `u` to `v` in
              the Hasse diagram, and this chain is lexicographically
              first among all chains from `u` to `v`.

        For more details, see [Bj1980]_.

        INPUT:

        - ``f`` -- a function taking two elements ``a`` and ``b`` in
          ``self`` such that ``b`` covers ``a`` and returning elements
          in a totally ordered set.

        - ``return_raising_chains`` (optional; default:``False``) if
          ``True``, returns the set of all raising chains in ``self``,
          if possible.

        EXAMPLES:

        Let us consider a Boolean poset::

            sage: P = Poset([[(0,0),(0,1),(1,0),(1,1)],[[(0,0),(0,1)],[(0,0),(1,0)],[(0,1),(1,1)],[(1,0),(1,1)]]],facade=True)
            sage: label = lambda a,b: min( i for i in [0,1] if a[i] != b[i] )
            sage: P.is_EL_labelling(label)
            True
            sage: P.is_EL_labelling(label,return_raising_chains=True)
            {((0, 0), (0, 1)): [1],
             ((0, 0), (1, 0)): [0],
             ((0, 0), (1, 1)): [0, 1],
             ((0, 1), (1, 1)): [0],
             ((1, 0), (1, 1)): [1]}
        """
        label_dict = {(a, b): f(a, b)
                      for a, b in self.cover_relations_iterator()}
        if return_raising_chains:
            raising_chains = {}
        for a, b in self.relations_iterator(strict=True):
            P = self.subposet(self.interval(a, b))
            max_chains = sorted([[label_dict[(chain[i], chain[i + 1])]
                                  for i in range(len(chain) - 1)]
                                 for chain in P.maximal_chains_iterator()])
            if max_chains[0] != sorted(max_chains[0]) or any(max_chains[i] == sorted(max_chains[i]) for i in range(1, len(max_chains))):
                return False
            elif return_raising_chains:
                raising_chains[(a, b)] = max_chains[0]
        if return_raising_chains:
            return raising_chains
        else:
            return True

    def dimension(self, certificate=False, *, solver=None, integrality_tolerance=1e-3):
        r"""
        Return the dimension of the Poset.

        The (Dushnik-Miller) dimension of a poset is the minimal
        number of total orders so that the poset is their
        "intersection".  More precisely, the dimension of a poset
        defined on a set `X` of points is the smallest integer `n`
        such that there exist linear extensions `P_1,...,P_n` of `P`
        satisfying:

        .. MATH::

            u\leq_P v\ \text{if and only if }\ \forall i, u\leq_{P_i} v

        For more information, see the :wikipedia:`Order_dimension`.

        INPUT:

        - ``certificate`` (boolean; default:``False``) -- whether to return an
          integer (the dimension) or a certificate, i.e. a smallest set of
          linear extensions.

        - ``solver`` -- (default: ``None``) Specify a Mixed Integer Linear Programming
          (MILP) solver to be used. If set to ``None``, the default one is used. For
          more information on MILP solvers and which default solver is used, see
          the method
          :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>`
          of the class
          :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.

        - ``integrality_tolerance`` -- parameter for use with MILP solvers over an
          inexact base ring; see :meth:`MixedIntegerLinearProgram.get_values`.

        .. NOTE::

            The speed of this function greatly improves when more efficient
            MILP solvers (e.g. Gurobi, CPLEX) are installed. See
            :class:`MixedIntegerLinearProgram` for more information.

        .. NOTE::

            Prior to version 8.3 this returned only realizer with
            ``certificate=True``. Now it returns a pair having a realizer as
            the second element. See :trac:`25588` for details.

        ALGORITHM:

        As explained [FT00]_, the dimension of a poset is equal to the (weak)
        chromatic number of a hypergraph. More precisely:

            Let `inc(P)` be the set of (ordered) pairs of incomparable elements
            of `P`, i.e. all `uv` and `vu` such that `u\not \leq_P v` and `v\not
            \leq_P u`. Any linear extension of `P` is a total order on `X` that
            can be seen as the union of relations from `P` along with some
            relations from `inc(P)`. Thus, the dimension of `P` is the smallest
            number of linear extensions of `P` which *cover* all points of
            `inc(P)`.

            Consequently, `dim(P)` is equal to the chromatic number of the
            hypergraph `\mathcal H_{inc}`, where `\mathcal H_{inc}` is the
            hypergraph defined on `inc(P)` whose sets are all `S\subseteq
            inc(P)` such that `P\cup S` is not acyclic.

        We solve this problem through a :mod:`Mixed Integer Linear Program
        <sage.numerical.mip>`.

        The problem is known to be NP-complete.

        EXAMPLES:

        We create a poset, compute a set of linear extensions and check
        that we get back the poset from them::

            sage: P = Poset([[1,4], [3], [4,5,3], [6], [], [6], []])
            sage: P.dimension()
            3
            sage: dim, L = P.dimension(certificate=True)
            sage: L  # random -- architecture-dependent
            [[0, 2, 4, 5, 1, 3, 6], [2, 5, 0, 1, 3, 4, 6], [0, 1, 2, 3, 5, 6, 4]]
            sage: Poset( (L[0], lambda x, y: all(l.index(x) < l.index(y) for l in L)) ) == P
            True

        According to Schnyder's theorem, the incidence poset (of
        height 2) of a graph has dimension `\leq 3` if and only if
        the graph is planar::

            sage: G = graphs.CompleteGraph(4)
            sage: P = Poset(DiGraph({(u,v):[u,v] for u,v,_ in G.edges()}))
            sage: P.dimension()
            3

            sage: G = graphs.CompleteBipartiteGraph(3,3)
            sage: P = Poset(DiGraph({(u,v):[u,v] for u,v,_ in G.edges()}))
            sage: P.dimension() # not tested - around 4s with CPLEX
            4

        TESTS:

        Empty Poset::

            sage: Poset().dimension()
            0
            sage: Poset().dimension(certificate=True)
            (0, [])

        Chain and certificate, :trac:`26861`::

            sage: Poset({'a': ['b']}).dimension(certificate=True)[1]
            [['a', 'b']]
        """
        if self.cardinality() == 0:
            return (0, []) if certificate else 0
        if self.is_chain():
            return (1, [self.list()]) if certificate else 1

        # current bound on the chromatic number of the hypergraph
        k = 2
        # if a realizer is not needed, we can optimize a little
        if not certificate:
            # polynomial time check for dimension 2
            from sage.graphs.comparability import greedy_is_comparability as is_comparability
            if is_comparability(self._hasse_diagram.transitive_closure().to_undirected().complement()):
                return 2
            k = 3
            # known upper bound for dimension
            max_value = max(self.cardinality() // 2, self.width())

        from sage.numerical.mip import MixedIntegerLinearProgram, MIPSolverException
        P = Poset(self._hasse_diagram)  # work on an int-labelled poset
        hasse_diagram = P.hasse_diagram()
        inc_graph = P.incomparability_graph()
        inc_P = inc_graph.edges(labels=False)

        # cycles is the list of all cycles found during the execution of the
        # algorithm

        cycles = [[(u, v), (v, u)] for u, v in inc_P]

        def init_LP(k, cycles, inc_P):
            r"""
            Initialize a LP object with k colors and the constraints from 'cycles'

                sage: init_LP(1,2,3) # not tested
            """
            p = MixedIntegerLinearProgram(constraint_generation=True, solver=solver)
            b = p.new_variable(binary=True)
            for (u, v) in inc_P:  # Each point has a color
                p.add_constraint(p.sum(b[(u, v), i] for i in range(k)) == 1)
                p.add_constraint(p.sum(b[(v, u), i] for i in range(k)) == 1)
            for cycle in cycles:  # No monochromatic set
                for i in range(k):
                    p.add_constraint(p.sum(b[point, i] for point in cycle) <= len(cycle) - 1)
            return p, b

        p, b = init_LP(k, cycles, inc_P)

        while True:
            if not certificate and k == max_value:
                return k
            # Compute a coloring of the hypergraph. If there is a problem,
            # increase the number of colors and start again.
            try:
                p.solve()
            except MIPSolverException:
                k += 1
                p, b = init_LP(k, cycles, inc_P)
                continue

            # We create the digraphs of all color classes
            linear_extensions = [hasse_diagram.copy() for i in range(k)]
            for ((u, v), i), x in p.get_values(b, convert=bool, tolerance=integrality_tolerance).items():
                if x:
                    linear_extensions[i].add_edge(u, v)

            # We check that all color classes induce an acyclic graph, and add a
            # constraint otherwise.
            okay = True
            for g in linear_extensions:
                is_acyclic, cycle = g.is_directed_acyclic(certificate=True)
                if not is_acyclic:
                    okay = False  # one is not acyclic
                    cycle = [(cycle[i - 1], cycle[i]) for i in range(len(cycle))]
                    cycle = [(u, v) for u, v in cycle if not P.lt(u, v) and not P.lt(v, u)]
                    cycles.append(cycle)
                    for i in range(k):
                        p.add_constraint(p.sum(b[point, i] for point in cycle) <= len(cycle) - 1)
            if okay:
                break

        linear_extensions = [g.topological_sort() for g in linear_extensions]

        if certificate:
            return (k, [[self._list[i] for i in l]
                    for l in linear_extensions])
        return k

    def jump_number(self, certificate=False):
        r"""
        Return the jump number of the poset.

        A *jump* in a linear extension `[e_1, \ldots, e_n]` of a poset `P`
        is a pair `(e_i, e_{i+1})` so that `e_{i+1}` does not cover `e_i`
        in `P`. The jump number of a poset is the minimal number of jumps
        in linear extensions of a poset.

        INPUT:

        - ``certificate`` -- (default: ``False``) Whether to return
          a certificate

        OUTPUT:

        - If ``certificate=True`` return a pair `(n, l)` where
          `n` is the jump number and `l` is a linear extension
          with `n` jumps. If ``certificate=False`` return only
          the jump number.

        EXAMPLES::

            sage: B3 = posets.BooleanLattice(3)
            sage: B3.jump_number()
            3

            sage: N = Poset({1: [3, 4], 2: [3]})
            sage: N.jump_number(certificate=True)
            (1, [1, 4, 2, 3])

        ALGORITHM:

        It is known that every poset has a greedy linear extension --
        an extension `[e_1, e_2, \ldots, e_n]` where every `e_{i+1}` is
        an upper cover of `e_i` if that is possible -- with the smallest
        possible number of jumps; see [Sys1987]_.

        Hence it suffices to test only those. We do that by backtracking.

        The problem is proven to be NP-complete.

        .. SEEALSO:: :meth:`is_jump_critical`

        TESTS::

            sage: E = Poset()
            sage: E.jump_number(certificate=True)
            (0, [])

            sage: C4 = posets.ChainPoset(4)
            sage: A4 = posets.AntichainPoset(4)
            sage: C4.jump_number()
            0
            sage: A4.jump_number()
            3
        """
        N = self.cardinality()
        if N == 0:
            return (0, []) if certificate else 0
        self_as_set = set(self)
        # No 'nonlocal' in Python 2. Workaround:
        # nonlocals[0] is the best jump number found so far,
        # nonlocals[1] is the linear extension giving it.
        nonlocals = [N, []]

        def greedy_rec(self, linext, jumpcount):
            """
            Recursively extend beginning of a linear extension by one element,
            unless we see that an extension with smaller jump number already
            has been found.
            """
            if len(linext) == N:
                nonlocals[0] = jumpcount
                nonlocals[1] = linext[:]
                return

            S = []
            if linext:
                # S is elements where we can grow the chain without a jump.
                S = [x for x in self.upper_covers(linext[-1]) if
                     all(low in linext for low in self.lower_covers(x))]
            if not S:
                if jumpcount >= nonlocals[0] - 1:
                    return
                jumpcount += 1
                # S is minimal elements of the poset without elements in linext
                S_ = self_as_set.difference(set(linext))
                S = [x for x in S_ if
                     not any(low in S_ for low in self.lower_covers(x))]

            for e in S:
                greedy_rec(self, linext + [e], jumpcount)

        greedy_rec(self, [], -1)

        if certificate:
            return (nonlocals[0], nonlocals[1])
        return nonlocals[0]

    def is_jump_critical(self, certificate=False):
        """
        Return ``True`` if the poset is jump-critical, and ``False`` otherwise.

        A poset `P` is *jump-critical* if every proper subposet has smaller
        jump number.

        INPUT:

        - ``certificate`` -- (default: ``False``) whether to return
          a certificate

        OUTPUT:

        - If ``certificate=True`` return either ``(True, None)`` or
          ``(False, e)`` so that removing element `e` from the poset does not
          decrease the jump number. If ``certificate=False`` return
          ``True`` or ``False``.

        EXAMPLES::

            sage: P = Poset({1: [3, 6], 2: [3, 4, 5], 4: [6, 7], 5: [7]})
            sage: P.is_jump_critical()
            True

            sage: P = posets.PentagonPoset()
            sage: P.is_jump_critical()
            False
            sage: P.is_jump_critical(certificate=True)
            (False, 3)

        .. SEEALSO:: :meth:`jump_number`

        TESTS::

            sage: Poset().is_jump_critical()
            True
            sage: Poset().is_jump_critical(certificate=True)
            (True, None)
        """
        # Quick check first
        for e in self:
            up = self.upper_covers(e)
            if len(up) == 1:
                if len(self.upper_covers(up[0])) == 1:
                    if certificate:
                        return (False, up[0])
                    return False
        # And now just dumb algorithm
        jumps = self.jump_number()
        for e in self:
            P = self.subposet([x for x in self if x != e])
            if P.jump_number() == jumps:
                return (False, e) if certificate else False
        return (True, None) if certificate else True

    def rank_function(self):
        r"""
        Return the (normalized) rank function of the poset,
        if it exists.

        A *rank function* of a poset `P` is a function `r`
        that maps elements of `P` to integers and satisfies:
        `r(x) = r(y) + 1` if `x` covers `y`. The function `r`
        is normalized such that its minimum value on every
        connected component of the Hasse diagram of `P` is
        `0`. This determines the function `r` uniquely (when
        it exists).

        OUTPUT:

        - a lambda function, if the poset admits a rank function
        - ``None``, if the poset does not admit a rank function

        EXAMPLES::

            sage: P = Poset(([1,2,3,4],[[1,4],[2,3],[3,4]]), facade=True)
            sage: P.rank_function() is not None
            True
            sage: P = Poset(([1,2,3,4,5],[[1,2],[2,3],[3,4],[1,5],[5,4]]), facade=True)
            sage: P.rank_function() is not None
            False
            sage: P = Poset(([1,2,3,4,5,6,7,8],[[1,4],[2,3],[3,4],[5,7],[6,7]]), facade=True)
            sage: f = P.rank_function(); f is not None
            True
            sage: f(5)
            0
            sage: f(2)
            0

        TESTS::

            sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]])
            sage: r = P.rank_function()
            sage: for u,v in P.cover_relations_iterator():
            ....:     if r(v) != r(u) + 1:
            ....:         print("Bug in rank_function!")

        ::

            sage: Q = Poset([[1,2],[4],[3],[4],[]])
            sage: Q.rank_function() is None
            True
        """
        hasse_rf = self._hasse_diagram.rank_function()
        if hasse_rf is None:
            return None
        else:
            return lambda z: hasse_rf(self._element_to_vertex(z))

    def rank(self, element=None):
        r"""
        Return the rank of an element ``element`` in the poset ``self``,
        or the rank of the poset if ``element`` is ``None``.

        (The rank of a poset is the length of the longest chain of
        elements of the poset. This is sometimes called the length of a poset.)

        EXAMPLES::

            sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]], facade = False)
            sage: P.rank(5)
            2
            sage: P.rank()
            3
            sage: Q = Poset([[1,2],[3],[],[]])

            sage: P = posets.SymmetricGroupBruhatOrderPoset(4)
            sage: [(v,P.rank(v)) for v in P]
            [('1234', 0),
             ('1243', 1),
            ...
             ('4312', 5),
             ('4321', 6)]
        """
        if element is None:
            return len(self.level_sets()) - 1
        elif self.is_ranked():
            return self.rank_function()(element)
        else:
            raise ValueError("the poset is not ranked")

    def is_ranked(self) -> bool:
        r"""
        Return ``True`` if the poset is ranked, and ``False`` otherwise.

        A poset is ranked if there is a function `r` from  poset elements
        to integers so that `r(x)=r(y)+1` when `x` covers `y`.

        Informally said a ranked poset can be "levelized": every element is
        on a "level", and every cover relation goes only one level up.

        EXAMPLES::

            sage: P = Poset( ([1, 2, 3, 4], [[1, 2], [2, 4], [3, 4]] ))
            sage: P.is_ranked()
            True

            sage: P = Poset([[1, 5], [2, 6], [3], [4],[], [6, 3], [4]])
            sage: P.is_ranked()
            False

        .. SEEALSO:: :meth:`rank_function`, :meth:`rank`, :meth:`is_graded`

        TESTS::

            sage: Poset().is_ranked()  # Test empty poset
            True
        """
        return bool(self.rank_function())

    def is_graded(self) -> bool:
        r"""
        Return ``True`` if the poset is graded, and ``False`` otherwise.

        A poset is graded if all its maximal chains have the same length.

        There are various competing definitions for graded
        posets (see :wikipedia:`Graded_poset`). This definition is from
        section 3.1 of Richard Stanley's *Enumerative Combinatorics,
        Vol. 1* [EnumComb1]_. Some sources call these posets *tiered*.

        Every graded poset is ranked. The converse is true
        for bounded posets, including lattices.

        EXAMPLES::

            sage: P = posets.PentagonPoset()  # Not even ranked
            sage: P.is_graded()
            False

            sage: P = Poset({1:[2, 3], 3:[4]})  # Ranked, but not graded
            sage: P.is_graded()
            False

            sage: P = Poset({1:[3, 4], 2:[3, 4], 5:[6]})
            sage: P.is_graded()
            True

            sage: P = Poset([[1], [2], [], [4], []])
            sage: P.is_graded()
            False

        .. SEEALSO:: :meth:`is_ranked`, :meth:`level_sets`

        TESTS::

            sage: Poset().is_graded()  # Test empty poset
            True
        """
        if len(self) <= 2:
            return True
        # Let's work with the Hasse diagram in order to avoid some
        # indirection (the output does not depend on the vertex labels).
        hasse = self._hasse_diagram
        rf = hasse.rank_function()
        if rf is None:
            return False    # because every graded poset is ranked.
        if not all(rf(i) == 0 for i in hasse.minimal_elements()):
            return False
        maxes = hasse.maximal_elements()
        rank = rf(maxes[0])
        return all(rf(i) == rank for i in maxes)

    def covers(self, x, y):
        """
        Return ``True`` if ``y`` covers ``x`` and ``False`` otherwise.

        Element `y` covers `x` if `x < y` and there is no `z` such that
        `x < z < y`.

        EXAMPLES::

            sage: P = Poset([[1,5], [2,6], [3], [4], [], [6,3], [4]])
            sage: P.covers(1, 6)
            True
            sage: P.covers(1, 4)
            False
            sage: P.covers(1, 5)
            False
        """
        return self._hasse_diagram.has_edge(*[self._element_to_vertex(w) for w in (x, y)])

    def upper_covers_iterator(self, x):
        """
        Return an iterator over the upper covers of the element ``x``.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[]})
            sage: type(P.upper_covers_iterator(0))
            <... 'generator'>
        """
        for e in self._hasse_diagram.neighbor_out_iterator(self._element_to_vertex(x)):
            yield self._vertex_to_element(e)

    def upper_covers(self, x):
        """
        Return the list of upper covers of the element ``x``.

        An upper cover of `x` is an element `y` such that `x < y` and
        there is no element `z` so that `x < z < y`.

        EXAMPLES::

            sage: P = Poset([[1,5], [2,6], [3], [4], [], [6,3], [4]])
            sage: P.upper_covers(1)
            [2, 6]

        .. SEEALSO:: :meth:`lower_covers`
        """
        return [e for e in self.upper_covers_iterator(x)]

    def lower_covers_iterator(self, x):
        """
        Return an iterator over the lower covers of the element ``x``.

        EXAMPLES::

            sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[]})
            sage: l0 = P.lower_covers_iterator(3)
            sage: type(l0)
            <... 'generator'>
            sage: next(l0)
            2
        """
        for e in self._hasse_diagram.neighbor_in_iterator(self._element_to_vertex(x)):
            yield self._vertex_to_element(e)

    def lower_covers(self, x):
        """
        Return the list of lower covers of the element ``x``.

        A lower cover of `x` is an element `y` such that `y < x` and
        there is no element `z` so that `y < z < x`.

        EXAMPLES::

            sage: P = Poset([[1,5], [2,6], [3], [4], [], [6,3], [4]])
            sage: P.lower_covers(3)
            [2, 5]
            sage: P.lower_covers(0)
            []

        .. SEEALSO:: :meth:`upper_covers`
        """
        return [e for e in self.lower_covers_iterator(x)]

    def cardinality(self):
        """
        Return the number of elements in the poset.

        EXAMPLES::

            sage: Poset([[1,2,3],[4],[4],[4],[]]).cardinality()
            5

        .. SEEALSO::

            :meth:`degree_polynomial` for a more refined invariant
        """
        return Integer(self._hasse_diagram.order())

    def moebius_function(self, x, y):
        r"""
        Return the value of the Möbius function of the poset on the
        elements x and y.

        EXAMPLES::

            sage: P = Poset([[1,2,3],[4],[4],[4],[]])
            sage: P.moebius_function(P(0),P(4))
            2
            sage: sum(P.moebius_function(P(0),v) for v in P)
            0
            sage: sum(abs(P.moebius_function(P(0),v))
            ....:     for v in P)
            6
            sage: for u,v in P.cover_relations_iterator():
            ....:     if P.moebius_function(u,v) != -1:
            ....:         print("Bug in moebius_function!")

        ::

            sage: Q = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]])
            sage: Q.moebius_function(Q(0),Q(7))
            0
            sage: Q.moebius_function(Q(0),Q(5))
            0
            sage: Q.moebius_function(Q(2),Q(7))
            2
            sage: Q.moebius_function(Q(3),Q(3))
            1
            sage: sum([Q.moebius_function(Q(0),v) for v in Q])
            0
        """
        i, j = map(self._element_to_vertex, (x, y))
        return self._hasse_diagram.moebius_function(i, j)

    def moebius_function_matrix(self, ring=ZZ, sparse=False):
        r"""
        Return a matrix whose ``(i,j)`` entry is the value of the Möbius
        function evaluated at ``self.linear_extension()[i]`` and
        ``self.linear_extension()[j]``.

        INPUT:

        - ``ring`` -- the ring of coefficients (default: ``ZZ``)
        - ``sparse`` -- whether the returned matrix is sparse or not
          (default: ``True``)

        EXAMPLES::

            sage: P = Poset([[4,2,3],[],[1],[1],[1]])
            sage: x,y = (P.linear_extension()[0],P.linear_extension()[1])
            sage: P.moebius_function(x,y)
            -1
            sage: M = P.moebius_function_matrix(); M
            [ 1 -1 -1 -1  2]
            [ 0  1  0  0 -1]
            [ 0  0  1  0 -1]
            [ 0  0  0  1 -1]
            [ 0  0  0  0  1]
            sage: M[0,4]
            2
            sage: M[0,1]
            -1

        We now demonstrate the usage of the optional parameters::

            sage: P.moebius_function_matrix(ring=QQ, sparse=False).parent()
            Full MatrixSpace of 5 by 5 dense matrices over Rational Field
        """
        M = self._hasse_diagram.moebius_function_matrix()
        if ring is not ZZ:
            M = M.change_ring(ring)
        if not sparse:
            M = M.dense_matrix()
        return M

    def lequal_matrix(self, ring=ZZ, sparse=False):
        """
        Compute the matrix whose ``(i,j)`` entry is 1 if
        ``self.linear_extension()[i] < self.linear_extension()[j]`` and 0
        otherwise.

        INPUT:

        - ``ring`` -- the ring of coefficients (default: ``ZZ``)
        - ``sparse`` -- whether the returned matrix is sparse or not
          (default: ``True``)

        EXAMPLES::

            sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]], facade = False)
            sage: LEQM = P.lequal_matrix(); LEQM
            [1 1 1 1 1 1 1 1]
            [0 1 0 1 0 0 0 1]
            [0 0 1 1 1 0 1 1]
            [0 0 0 1 0 0 0 1]
            [0 0 0 0 1 0 0 1]
            [0 0 0 0 0 1 1 1]
            [0 0 0 0 0 0 1 1]
            [0 0 0 0 0 0 0 1]
            sage: LEQM[1,3]
            1
            sage: P.linear_extension()[1] < P.linear_extension()[3]
            True
            sage: LEQM[2,5]
            0
            sage: P.linear_extension()[2] < P.linear_extension()[5]
            False

        We now demonstrate the usage of the optional parameters::

            sage: P.lequal_matrix(ring=QQ, sparse=False).parent()
            Full MatrixSpace of 8 by 8 dense matrices over Rational Field
        """
        M = self._hasse_diagram.lequal_matrix(boolean=False)
        if ring is not ZZ:
            M = M.change_ring(ring)
        if not sparse:
            M = M.dense_matrix()
        return M

    def coxeter_transformation(self):
        r"""
        Return the Coxeter transformation of the poset.

        OUTPUT:

        a square matrix with integer coefficients

        The output is the matrix of the Auslander-Reiten translation
        acting on the Grothendieck group of the derived category of
        modules on the poset, in the basis of simple
        modules. This matrix is usually called the Coxeter
        transformation.

        EXAMPLES::

            sage: posets.PentagonPoset().coxeter_transformation()
            [ 0  0  0  0 -1]
            [ 0  0  0  1 -1]
            [ 0  1  0  0 -1]
            [-1  1  1  0 -1]
            [-1  1  0  1 -1]

        .. SEEALSO::

            :meth:`coxeter_polynomial`, :meth:`coxeter_smith_form`

        TESTS::

            sage: M = posets.PentagonPoset().coxeter_transformation()
            sage: M ** 8 == 1
            True
        """
        return self._hasse_diagram.coxeter_transformation()

    def coxeter_polynomial(self):
        """
        Return the Coxeter polynomial of the poset.

        OUTPUT:

        a polynomial in one variable

        The output is the characteristic polynomial of the Coxeter
        transformation. This polynomial only depends on the derived
        category of modules on the poset.

        EXAMPLES::

            sage: P = posets.PentagonPoset()
            sage: P.coxeter_polynomial()
            x^5 + x^4 + x + 1

            sage: p = posets.SymmetricGroupWeakOrderPoset(3)
            sage: p.coxeter_polynomial()
            x^6 + x^5 - x^3 + x + 1

        .. SEEALSO::

            :meth:`coxeter_transformation`, :meth:`coxeter_smith_form`
        """
        return self._hasse_diagram.coxeter_transformation().charpoly()

    def coxeter_smith_form(self, algorithm='singular'):
        """
        Return the Smith normal form of `x` minus the Coxeter transformation
        matrix.

        INPUT:

        - ``algorithm`` -- optional (default ``'singular'``), possible
          values are ``'singular'``, ``'sage'``, ``'gap'``,
          ``'pari'``, ``'maple'``, ``'magma'``, ``'fricas'``

        Beware that speed depends very much on the choice of
        algorithm. Sage is rather slow, Singular is faster and Pari is
        fast at least for small sizes.

        OUTPUT:

        - list of polynomials in one variable, each one dividing the next one

        The output list is a refinement of the characteristic polynomial of
        the Coxeter transformation, which is its product. This list
        of polynomials only depends on the derived category of modules
        on the poset.

        EXAMPLES::

           sage: P = posets.PentagonPoset()
           sage: P.coxeter_smith_form()
           [1, 1, 1, 1, x^5 + x^4 + x + 1]

           sage: P = posets.DiamondPoset(7)
           sage: prod(P.coxeter_smith_form()) == P.coxeter_polynomial()
           True

        TESTS::

           sage: P = posets.PentagonPoset()
           sage: P.coxeter_smith_form(algorithm='sage')
           [1, 1, 1, 1, x^5 + x^4 + x + 1]
           sage: P.coxeter_smith_form(algorithm='gap')
           [1, 1, 1, 1, x^5 + x^4 + x + 1]
           sage: P.coxeter_smith_form(algorithm='pari')
           [1, 1, 1, 1, x^5 + x^4 + x + 1]
           sage: P.coxeter_smith_form(algorithm='fricas')  # optional - fricas
           [1, 1, 1, 1, x^5 + x^4 + x + 1]
           sage: P.coxeter_smith_form(algorithm='maple')  # optional - maple
           [1, 1, 1, 1, x^5 + x^4 + x + 1]
           sage: P.coxeter_smith_form(algorithm='magma')  # optional - magma
           [1, 1, 1, 1, x^5 + x^4 + x + 1]

        .. SEEALSO::

            :meth:`coxeter_transformation`, :meth:`coxeter_matrix`
        """
        c0 = self.coxeter_transformation()
        x = polygen(QQ, 'x')   # not possible to use ZZ for the moment

        if algorithm == 'singular':  # quite faster than sage
            from sage.interfaces.singular import singular
            singular.LIB('jacobson.lib')
            sing_m = singular(x - c0)
            L = sing_m.smith().sage().diagonal()
            return sorted([u / u.lc() for u in L],
                          key=lambda p: p.degree())

        if algorithm == 'sage':  # *very slow*
            return (x - c0).smith_form(transformation=False).diagonal()

        if algorithm == 'magma':  # also quite fast
            from sage.interfaces.magma import magma
            elem = magma('ElementaryDivisors')
            return elem.evaluate(x - c0).sage()

        if algorithm == 'gap':
            from sage.libs.gap.libgap import libgap
            gap_m = libgap(x - c0)
            elem = gap_m.ElementaryDivisorsMat()
            return elem.sage()

        if algorithm == 'pari':   # maybe fast, at least for small size
            from sage.libs.pari import pari
            pari_m = pari(x - c0)
            elem = pari_m.matsnf(2)
            A = x.parent()
            return sorted((A(f) for f in elem),
                          key=lambda p: p.degree())

        if algorithm == 'maple':
            from sage.interfaces.maple import maple
            maple_m = maple(x - c0)
            maple.load("MatrixPolynomialAlgebra")
            maple.load("ArrayTools")
            elem = maple.SmithForm(maple_m).Diagonal()
            A = x.parent()
            return [A(f.sage()) for f in elem]

        if algorithm == 'fricas':
            from sage.interfaces.fricas import fricas
            fm = fricas(x - c0)
            return list(fricas(fm.name() + "::Matrix(UP(x, FRAC INT))").smith().diagonal().sage())

    def is_meet_semilattice(self, certificate=False):
        r"""
        Return ``True`` if the poset has a meet operation, and
        ``False`` otherwise.

        A meet is the greatest lower bound for given elements, if it exists.

        INPUT:

        - ``certificate`` -- (default: ``False``) whether to return
          a certificate

        OUTPUT:

        - If ``certificate=True`` return either ``(True, None)`` or
          ``(False, (a, b))`` where elements `a` and `b` have no
          greatest lower bound. If ``certificate=False`` return
          ``True`` or ``False``.

        EXAMPLES::

            sage: P = Poset({1:[2, 3, 4], 2:[5, 6], 3:[6], 4:[6, 7]})
            sage: P.is_meet_semilattice()
            True

            sage: Q = P.dual()
            sage: Q.is_meet_semilattice()
            False

            sage: V = posets.IntegerPartitions(5)
            sage: V.is_meet_semilattice(certificate=True)
            (False, ((2, 2, 1), (3, 1, 1)))

        .. SEEALSO::

            - Dual property: :meth:`is_join_semilattice`
            - Stronger properties: :meth:`~sage.categories.finite_posets.FinitePosets.ParentMethods.is_lattice`

        TESTS::

            sage: Poset().is_meet_semilattice()  # Test empty lattice
            True
            sage: len([P for P in Posets(4) if P.is_meet_semilattice()])
            5

            sage: P = Poset({1: [2], 3: []})
            sage: P.is_meet_semilattice(certificate=True)
            (False, (3, 1))
        """
        from sage.combinat.posets.hasse_diagram import LatticeError
        try:
            self._hasse_diagram.meet_matrix()
        except LatticeError as error:
            if not certificate:
                return False
            x = self._vertex_to_element(error.x)
            y = self._vertex_to_element(error.y)
            return (False, (x, y))
        except ValueError as error:
            if error.args[0] != 'not a meet-semilattice: no bottom element':
                raise
            if not certificate:
                return False
            i = 1
            while self._hasse_diagram.in_degree(i) > 0:
                i += 1
            x = self._vertex_to_element(0)
            y = self._vertex_to_element(i)
            return (False, (x, y))
        if certificate:
            return (True, None)
        return True

    def is_join_semilattice(self, certificate=False):
        """
        Return ``True`` if the poset has a join operation, and ``False``
        otherwise.

        A join is the least upper bound for given elements, if it exists.

        INPUT:

        - ``certificate`` -- (default: ``False``) whether to return
          a certificate

        OUTPUT:

        - If ``certificate=True`` return either ``(True, None)`` or
          ``(False, (a, b))`` where elements `a` and `b` have no
          least upper bound. If ``certificate=False`` return
          ``True`` or ``False``.

        EXAMPLES::

            sage: P = Poset([[1,3,2], [4], [4,5,6], [6], [7], [7], [7], []])
            sage: P.is_join_semilattice()
            True

            sage: P = Poset({1:[3, 4], 2:[3, 4], 3:[5], 4:[5]})
            sage: P.is_join_semilattice()
            False
            sage: P.is_join_semilattice(certificate=True)
            (False, (2, 1))

        .. SEEALSO::

            - Dual property: :meth:`is_meet_semilattice`
            - Stronger properties: :meth:`~sage.categories.finite_posets.FinitePosets.ParentMethods.is_lattice`

        TESTS::

            sage: Poset().is_join_semilattice()  # Test empty lattice
            True
            sage: len([P for P in Posets(4) if P.is_join_semilattice()])
            5

            sage: X = Poset({1: [3], 2: [3], 3: [4, 5]})
            sage: X.is_join_semilattice(certificate=True)
            (False, (5, 4))
        """
        from sage.combinat.posets.hasse_diagram import LatticeError
        try:
            self._hasse_diagram.join_matrix()
        except LatticeError as error:
            if not certificate:
                return False
            x = self._vertex_to_element(error.x)
            y = self._vertex_to_element(error.y)
            return (False, (x, y))
        except ValueError as error:
            if error.args[0] != 'not a join-semilattice: no top element':
                raise
            if not certificate:
                return False
            n = self.cardinality() - 1
            i = n - 1
            while self._hasse_diagram.out_degree(i) > 0:
                i -= 1
            x = self._vertex_to_element(n)
            y = self._vertex_to_element(i)
            return (False, (x, y))
        if certificate:
            return (True, None)
        return True

    def is_isomorphic(self, other, **kwds):
        """
        Return ``True`` if both posets are isomorphic.

        EXAMPLES::

            sage: P = Poset(([1,2,3],[[1,3],[2,3]]))
            sage: Q = Poset(([4,5,6],[[4,6],[5,6]]))
            sage: P.is_isomorphic(Q)
            True

        TESTS:

        Since :trac:`25576`, one can ask for the isomorphism::

            sage: P.is_isomorphic(Q, certificate=True)
            (True, {1: 4, 2: 5, 3: 6})
        """
        if hasattr(other, 'hasse_diagram'):
            return self.hasse_diagram().is_isomorphic(other.hasse_diagram(),
                                                      **kwds)
        else:
            raise TypeError("'other' is not a finite poset")

    def isomorphic_subposets_iterator(self, other):
        """
        Return an iterator over the subposets of ``self`` isomorphic to
        ``other``.

        By subposet we mean ``self.subposet(X)`` which is isomorphic
        to ``other`` and where ``X`` is a subset of elements of
        ``self``.

        INPUT:

        - ``other`` -- a finite poset

        EXAMPLES::

            sage: D = Poset({1:[2,3], 2:[4], 3:[4]})
            sage: N5 = posets.PentagonPoset()
            sage: for P in N5.isomorphic_subposets_iterator(D):
            ....:     print(P.cover_relations())
            [[0, 1], [0, 2], [1, 4], [2, 4]]
            [[0, 1], [0, 3], [1, 4], [3, 4]]
            [[0, 1], [0, 2], [1, 4], [2, 4]]
            [[0, 1], [0, 3], [1, 4], [3, 4]]

        .. WARNING::

            This function will return same subposet as many times as
            there are automorphism on it. This is due to
            :meth:`~sage.graphs.generic_graph.GenericGraph.subgraph_search_iterator`
            returning labelled subgraphs. On the other hand, this
            function does not eat memory like
            :meth:`isomorphic_subposets` does.

        .. SEEALSO::

            :meth:`sage.combinat.posets.lattices.FiniteLatticePoset.isomorphic_sublattices_iterator`.
        """
        if not hasattr(other, 'hasse_diagram'):
            raise TypeError("'other' is not a finite poset")
        return (self.subposet([self._list[i] for i in x]) for x in self._hasse_diagram.transitive_closure().subgraph_search_iterator(other.hasse_diagram().transitive_closure(), induced=True))

    def isomorphic_subposets(self, other):
        """
        Return a list of subposets of ``self`` isomorphic to ``other``.

        By subposet we mean ``self.subposet(X)`` which is isomorphic to
        ``other`` and where ``X`` is a subset of elements of ``self``.

        INPUT:

        - ``other`` -- a finite poset

        EXAMPLES::

            sage: C2 = Poset({0:[1]})
            sage: C3 = Poset({'a':['b'], 'b':['c']})
            sage: L = sorted(x.cover_relations() for x in C3.isomorphic_subposets(C2))
            sage: for x in L: print(x)
            [['a', 'b']]
            [['a', 'c']]
            [['b', 'c']]

            sage: D = Poset({1:[2,3], 2:[4], 3:[4]})
            sage: N5 = posets.PentagonPoset()
            sage: len(N5.isomorphic_subposets(D))
            2

        .. NOTE::

            If this function takes too much time, try using
            :meth:`isomorphic_subposets_iterator`.
        """
        if not hasattr(other, 'hasse_diagram'):
            raise TypeError("'other' is not a finite poset")
        L = self._hasse_diagram.transitive_closure().subgraph_search_iterator(other._hasse_diagram.transitive_closure(), induced=True)
        # Since subgraph_search_iterator returns labelled copies, we
        # remove duplicates.
        return [self.subposet([self._list[i] for i in x]) for x in sorted(set(frozenset(y) for y in L))]

    # Caveat: list is overridden by the method list above!!!
    def antichains(self, element_constructor=type([])):
        """
        Return the antichains of the poset.

        An *antichain* of a poset is a set of elements of the
        poset that are pairwise incomparable.

        INPUT:

         - ``element_constructor`` -- a function taking an iterable as
           argument (default: ``list``)

        OUTPUT:

        The enumerated set (of type
        :class:`~sage.combinat.subsets_pairwise.PairwiseCompatibleSubsets`)
        of all antichains of the poset, each of which is given as an
        ``element_constructor.``

        EXAMPLES::

            sage: A = posets.PentagonPoset().antichains(); A
            Set of antichains of Finite lattice containing 5 elements
            sage: list(A)
            [[], [0], [1], [1, 2], [1, 3], [2], [3], [4]]
            sage: A.cardinality()
            8
            sage: A[3]
            [1, 2]

        To get the antichains as, say, sets, one may use the
        ``element_constructor`` option::

            sage: list(posets.ChainPoset(3).antichains(element_constructor=set))
            [set(), {0}, {1}, {2}]

        To get the antichains of a given size one can currently use::

            sage: list(A.elements_of_depth_iterator(2))
            [[1, 2], [1, 3]]

        Eventually the following syntax will be accepted::

            sage: A.subset(size = 2) # todo: not implemented

        .. NOTE::

            Internally, this uses
            :class:`sage.combinat.subsets_pairwise.PairwiseCompatibleSubsets`
            and :class:`RecursivelyEnumeratedSet_forest`. At this point, iterating
            through this set is about twice slower than using
            :meth:`antichains_iterator` (tested on
            ``posets.AntichainPoset(15)``). The algorithm is the same
            (depth first search through the tree), but
            :meth:`antichains_iterator` manually inlines things which
            apparently avoids some infrastructure overhead.

            On the other hand, this returns a full featured enumerated
            set, with containment testing, etc.

        .. SEEALSO:: :meth:`maximal_antichains`, :meth:`chains`
        """
        vertex_to_element = self._vertex_to_element

        def f(antichain):
            return element_constructor(vertex_to_element(x) for x in antichain)
        result = self._hasse_diagram.antichains(element_class=f)
        result.rename("Set of antichains of %s" % self)
        return result

    def antichains_iterator(self):
        """
        Return an iterator over the antichains of the poset.

        EXAMPLES::

            sage: it = posets.PentagonPoset().antichains_iterator(); it
            <generator object ...antichains_iterator at ...>
            sage: next(it), next(it)
            ([], [4])

        .. SEEALSO:: :meth:`antichains`
        """
        vertex_to_element = self._vertex_to_element
        for antichain in self._hasse_diagram.antichains_iterator():
            yield [vertex_to_element(_) for _ in antichain]

    def width(self, certificate=False):
        r"""
        Return the width of the poset (the size of its longest antichain).

        It is computed through a matching in a bipartite graph; see
        :wikipedia:`Dilworth%27s_theorem` for more information. The width is
        also called Dilworth number.

        INPUT:

        - ``certificate`` -- (default: ``False``) whether to return
          a certificate

        OUTPUT:

        - If ``certificate=True`` return ``(w, a)``, where `w` is the
          width of a poset and `a` is an antichain of maximum cardinality.
          If ``certificate=False`` return only width of the poset.

        EXAMPLES::

            sage: P = posets.BooleanLattice(4)
            sage: P.width()
            6

            sage: w, max_achain = P.width(certificate=True)
            sage: sorted(max_achain)
            [3, 5, 6, 9, 10, 12]

        TESTS::

            sage: Poset().width()
            0
            sage: Poset().width(certificate=True)
            (0, [])
        """
        if certificate:
            max_achain = self.incomparability_graph().clique_maximum()
            return (len(max_achain), max_achain)

        # See the doc of dilworth_decomposition for an explanation of what is
        # going on.
        from sage.graphs.graph import Graph
        n = self.cardinality()
        g = Graph()
        for v, u in self._hasse_diagram.transitive_closure().edge_iterator(labels=False):
            g.add_edge(u + n, v)
        return n - len(g.matching())

    def dilworth_decomposition(self):
        r"""
        Return a partition of the points into the minimal number of chains.

        According to Dilworth's theorem, the points of a poset can be
        partitioned into `\alpha` chains, where `\alpha` is the cardinality of
        its largest antichain. This method returns such a partition.

        See :wikipedia:`Dilworth%27s_theorem`.

        ALGORITHM:

        We build a bipartite graph in which a vertex `v` of the poset is
        represented by two vertices `v^-,v^+`. For any two `u,v` such that
        `u<v` in the poset we add an edge `v^+u^-`.

        A matching in this graph is equivalent to a partition of the poset
        into chains: indeed, a chain `v_1...v_k` gives rise to the matching
        `v_1^+v_2^-,v_2^+v_3^-,...`, and from a matching one can build the
        union of chains.

        According to Dilworth's theorem, the number of chains is equal to
        `\alpha` (the posets' width).

        EXAMPLES::

            sage: p = posets.BooleanLattice(4)
            sage: p.width()
            6
            sage: p.dilworth_decomposition()  # random
            [[7, 6, 4], [11, 3], [12, 8, 0], [13, 9, 1], [14, 10, 2], [15, 5]]


        .. SEEALSO::

            :meth:`level_sets` to return elements grouped to antichains.

        TESTS::

            sage: p = posets.IntegerCompositions(5)
            sage: d = p.dilworth_decomposition()
            sage: for chain in d:
            ....:    for i in range(len(chain)-1):
            ....:        assert p.is_greater_than(chain[i],chain[i+1])
            sage: set(p) == set().union(*d)
            True
        """
        from sage.graphs.graph import Graph
        n = self.cardinality()
        g = Graph()
        for v, u in self._hasse_diagram.transitive_closure().edge_iterator(labels=False):
            g.add_edge(u + n, v)
        matching = {}
        for u, v, _ in g.matching():
            matching[u] = v
            matching[v] = u
        chains = []
        for v in range(n):
            if v in matching:
                continue
            # v is the top element of its chain
            chain = []
            while True:
                chain.append(self._list[v])
                v = matching.get(v + n, None)
                if v is None:
                    break
            chains.append(chain)
        return chains

    def chains(self, element_constructor=type([]), exclude=None):
        """
        Return the chains of the poset.

        A *chain* of a poset is an increasing sequence of
        distinct elements of the poset.

        INPUT:

        - ``element_constructor`` -- a function taking an iterable as
          argument (optional, default: ``list``)

        - ``exclude`` -- elements of the poset to be excluded
          (optional, default: ``None``)

        OUTPUT:

        The enumerated set (of type
        :class:`~sage.combinat.subsets_pairwise.PairwiseCompatibleSubsets`)
        of all chains of the poset, each of which is given as an
        ``element_constructor``.

        EXAMPLES::

            sage: C = posets.PentagonPoset().chains(); C
            Set of chains of Finite lattice containing 5 elements
            sage: list(C)
            [[], [0], [0, 1], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], [1], [1, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]]

        Exclusion of elements, tuple (instead of list) as constructor::

            sage: P = Poset({1: [2, 3], 2: [4], 3: [4, 5]})
            sage: list(P.chains(element_constructor=tuple, exclude=[3]))
            [(), (1,), (1, 2), (1, 2, 4), (1, 4), (1, 5), (2,), (2, 4), (4,), (5,)]

        To get the chains of a given size one can currently use::

            sage: list(C.elements_of_depth_iterator(2))
            [[0, 1], [0, 2], [0, 3], [0, 4], [1, 4], [2, 3], [2, 4], [3, 4]]

        Eventually the following syntax will be accepted::

            sage: C.subset(size = 2) # todo: not implemented

        .. SEEALSO:: :meth:`maximal_chains`, :meth:`antichains`
        """
        if exclude is not None:
            exclude = [self._element_to_vertex(x) for x in exclude]
        result = self._hasse_diagram.chains(element_class=element_constructor,
                                            exclude=exclude,
                                            conversion=self._elements)
        result.rename("Set of chains of %s" % self)
        return result

    def connected_components(self):
        """
        Return the connected components of the poset as subposets.

        EXAMPLES::

            sage: P = Poset({1: [2, 3], 3: [4, 5], 6: [7, 8]})
            sage: parts = sorted(P.connected_components(), key=len); parts
            [Finite poset containing 3 elements,
             Finite poset containing 5 elements]
            sage: parts[0].cover_relations()
            [[6, 7], [6, 8]]

        .. SEEALSO:: :meth:`disjoint_union`, :meth:`is_connected`

        TESTS::

            sage: Poset().connected_components() # Test empty poset
            []

            sage: P = Poset({1: [2, 3], 3: [4, 5]})
            sage: CC = P.connected_components()
            sage: CC[0] is P
            True

            sage: P = Poset({1: [2, 3], 3: [4, 5], 6: [7, 8]}, facade=False)
            sage: V = sorted(P.connected_components(), key=len)[0]
            sage: V(7) < V(8)  # Facade argument should be inherited
            False
        """
        comps = self._hasse_diagram.connected_components_subgraphs()
        if len(comps) == 1:
            return [self]
        result = []
        if self._is_facade:
            for part in comps:
                G = part.relabel(self._vertex_to_element, inplace=False)
                result.append(Poset(G, cover_relations=True,
                                    facade=True))
        else:
            for part in comps:
                G = part.relabel(lambda v: self._vertex_to_element(v).element,
                                 inplace=False)
                result.append(Poset(G, cover_relations=True,
                                    facade=False))
        return result

    def ordinal_summands(self):
        r"""
        Return the ordinal summands of the poset as subposets.

        The ordinal summands of a poset `P` is the longest list of
        non-empty subposets `P_1, \ldots, P_n` whose ordinal sum is `P`. This
        decomposition is unique.

        EXAMPLES::

            sage: P = Poset({'a': ['c', 'd'], 'b': ['d'], 'c': ['x', 'y'],
            ....: 'd': ['x', 'y']})
            sage: parts = P.ordinal_summands(); parts
            [Finite poset containing 4 elements, Finite poset containing 2 elements]
            sage: sorted(parts[0])
            ['a', 'b', 'c', 'd']
            sage: Q = parts[0].ordinal_sum(parts[1])
            sage: Q.is_isomorphic(P)
            True

        .. SEEALSO::

            :meth:`ordinal_sum`

        ALGORITHM:

        Suppose that a poset `P` is the ordinal sum of posets `L` and `U`. Then
        `P` contains maximal antichains `l` and `u` such that every element of
        `u` covers every element of `l`; they correspond to maximal elements of
        `L` and minimal elements of `U`.

        We consider a linear extension `x_1,\ldots,x_n` of the poset's
        elements.

        We keep track of the maximal elements of subposet induced by elements
        `0,\ldots,x_i` and minimal elements of subposet induced by elements
        `x_{i+1},\ldots,x_n`, incrementing `i` one by one. We then check if
        `l` and `u` fit the previous description.

        TESTS::

            sage: Poset().ordinal_summands()
            [Finite poset containing 0 elements]
            sage: Poset({1: []}).ordinal_summands()
            [Finite poset containing 1 elements]
        """
        n = self.cardinality()
        if n <= 0:
            return [self]

        H = self._hasse_diagram
        cut_points = [-1]
        in_degrees = H.in_degree()
        lower = set([])
        upper = set(H.sources())

        for e in range(n):

            # update 'lower' by adding 'e' to it
            lower.add(e)
            lower.difference_update(H.neighbors_in(e))

            # update 'upper' too
            upper.discard(e)
            up_covers = H.neighbors_out(e)
            for uc in up_covers:
                in_degrees[uc] -= 1
                if in_degrees[uc] == 0:
                    upper.add(uc)

            if e + 1 in up_covers:
                uc_len = len(upper)
                for l in lower:
                    if H.out_degree(l) != uc_len:
                        break
                else:
                    for l in lower:
                        if set(H.neighbors_out(l)) != upper:
                            break
                    else:
                        cut_points.append(e)

        cut_points.append(n - 1)

        parts = []
        for i, j in zip(cut_points, cut_points[1:]):
            G = self._hasse_diagram.subgraph(range(i + 1, j + 1))
            parts.append(Poset(G.relabel(self._vertex_to_element,
                                         inplace=False)))
        return parts

    def product(self, other):
        r"""
        Return the Cartesian product of the poset with ``other``.

        The Cartesian (or 'direct') product of `P` and
        `Q` is defined by `(p, q) \le (p', q')` iff `p \le p'`
        in `P` and `q \le q'` in `Q`.

        Product of (semi)lattices are returned as a (semi)lattice.

        EXAMPLES::

            sage: P = posets.ChainPoset(3)
            sage: Q = posets.ChainPoset(4)
            sage: PQ = P.product(Q) ; PQ
            Finite lattice containing 12 elements
            sage: len(PQ.cover_relations())
            17
            sage: Q.product(P).is_isomorphic(PQ)
            True

            sage: P = posets.BooleanLattice(2)
            sage: Q = P.product(P)
            sage: Q.is_isomorphic(posets.BooleanLattice(4))
            True

        One can also simply use `*`::

            sage: P = posets.ChainPoset(2)
            sage: Q = posets.ChainPoset(3)
            sage: P*Q
            Finite lattice containing 6 elements

        .. SEEALSO::

            :class:`~sage.combinat.posets.cartesian_product.CartesianProductPoset`, :meth:`factor`

        TESTS::

            sage: Poset({0: [1]}).product(Poset())  # Product with empty poset
            Finite poset containing 0 elements
            sage: Poset().product(Poset())  # Product of two empty poset
            Finite poset containing 0 elements

        We check that :trac:`19113` is fixed::

            sage: L = LatticePoset({1: []})
            sage: type(L) == type(L.product(L))
            True
        """
        from sage.combinat.posets.lattices import LatticePoset, \
            JoinSemilattice, MeetSemilattice, FiniteLatticePoset, \
            FiniteMeetSemilattice, FiniteJoinSemilattice
        if (isinstance(self, FiniteLatticePoset) and
                isinstance(other, FiniteLatticePoset)):
            constructor = LatticePoset
        elif (isinstance(self, FiniteMeetSemilattice) and
              isinstance(other, FiniteMeetSemilattice)):
            constructor = MeetSemilattice
        elif (isinstance(self, FiniteJoinSemilattice) and
              isinstance(other, FiniteJoinSemilattice)):
            constructor = JoinSemilattice
        else:
            constructor = Poset
        return constructor(self.hasse_diagram().cartesian_product(other.hasse_diagram()))

    _mul_ = product

    def rees_product(self, other):
        r"""
        Return the Rees product of ``self`` and ``other``.

        This is only defined if both posets are graded.

        The underlying set is the set of pairs `(p,q)` in the Cartesian
        product such that `\operatorname{rk}(p) \geq \operatorname{rk}(q)`.

        This operation was defined by Björner and Welker in [BjWe2005]_.
        Other references are [MBRe2011]_ and [LSW2012]_.

        EXAMPLES::

            sage: B3 = posets.BooleanLattice(3)
            sage: B3t = B3.subposet(list(range(1,8)))
            sage: C3 = posets.ChainPoset(3)
            sage: D = B3t.rees_product(C3); D
            Finite poset containing 12 elements
            sage: sorted(D.minimal_elements())
            [(1, 0), (2, 0), (4, 0)]
            sage: sorted(D.maximal_elements())
            [(7, 0), (7, 1), (7, 2)]
            sage: D.with_bounds().moebius_function('bottom','top')
            2

        .. SEEALSO::

            :meth:`product`, :meth:`ordinal_product`, :meth:`star_product`

        TESTS::

            sage: T = posets.TamariLattice(3)
            sage: T.rees_product(T)
            Traceback (most recent call last):
            ...
            TypeError: the Rees product is defined only for graded posets
        """
        if not(self.is_graded() and other.is_graded()):
            raise TypeError('the Rees product is defined only for graded posets')

        rk0 = self.rank_function()
        rk1 = other.rank_function()
        rees_set = [(p, q) for p in self for q in other if rk0(p) >= rk1(q)]

        def covers(pq, ppqq):
            p, q = pq
            pp, qq = ppqq
            return self.covers(p, pp) and (q == qq or other.covers(q, qq))

        return Poset((rees_set, covers), cover_relations=True)

    def factor(self):
        """
        Factor the poset as a Cartesian product of smaller posets.

        This only works for connected posets for the moment.

        The decomposition of a connected poset as a Cartesian product
        of posets (prime in the sense that they cannot be written as
        Cartesian products) is unique up to reordering and
        isomorphism.

        OUTPUT:

        a list of posets

        EXAMPLES::

            sage: P = posets.PentagonPoset()
            sage: Q = P*P
            sage: Q.factor()
            [Finite poset containing 5 elements,
            Finite poset containing 5 elements]

            sage: P1 = posets.ChainPoset(3)
            sage: P2 = posets.ChainPoset(7)
            sage: P1.factor()
            [Finite lattice containing 3 elements]
            sage: (P1 * P2).factor()
            [Finite poset containing 7 elements,
            Finite poset containing 3 elements]

            sage: P = posets.TamariLattice(4)
            sage: (P*P).factor()
            [Finite poset containing 14 elements,
            Finite poset containing 14 elements]

        .. SEEALSO:: :meth:`product`

        TESTS::

            sage: P = posets.AntichainPoset(4)
            sage: P.factor()
            Traceback (most recent call last):
            ...
            NotImplementedError: the poset is not connected

            sage: P = posets.Crown(2)
            sage: P.factor()
            [Finite poset containing 4 elements]

            sage: Poset().factor()
            [Finite poset containing 0 elements]

            sage: factor(posets.BooleanLattice(2))
            [Finite poset containing 2 elements,
            Finite poset containing 2 elements]

        REFERENCES:

        .. [Feig1986] Joan Feigenbaum, *Directed Cartesian-Product Graphs
           have Unique Factorizations that can be computed in Polynomial Time*,
           Discrete Applied Mathematics 15 (1986) 105-110
           :doi:`10.1016/0166-218X(86)90023-5`
        """
        from sage.misc.mrange import cartesian_product_iterator
        from sage.graphs.graph import Graph
        from sage.misc.flatten import flatten
        dg = self._hasse_diagram
        if not dg.is_connected():
            raise NotImplementedError('the poset is not connected')
        if ZZ(dg.num_verts()).is_prime():
            return [self]
        G = dg.to_undirected()
        is_product, dic = G.is_cartesian_product(relabeling=True)
        if not is_product:
            return [self]
        dic = {key: tuple(flatten(dic[key])) for key in dic}

        prod_dg = dg.relabel(dic, inplace=False)
        v0 = next(iter(dic.values()))
        n = len(v0)
        factors_range = list(range(n))
        fusion = Graph(n)

        def edge_color(va, vb):
            for i in range(n):
                if va[i] != vb[i]:
                    return i

        for i0, i1 in Subsets(factors_range, 2):
            for x in prod_dg:
                neigh0 = [y for y in prod_dg.neighbors(x)
                          if edge_color(x, y) == i0]
                neigh1 = [z for z in prod_dg.neighbors(x)
                          if edge_color(x, z) == i1]
                for x0, x1 in cartesian_product_iterator([neigh0, neigh1]):
                    x2 = list(x0)
                    x2[i1] = x1[i1]
                    x2 = tuple(x2)
                    A0 = prod_dg.has_edge(x, x0)
                    B0 = prod_dg.has_edge(x1, x2)
                    if A0 != B0:
                        fusion.add_edge([i0, i1])
                        break
                    A1 = prod_dg.has_edge(x, x1)
                    B1 = prod_dg.has_edge(x0, x2)
                    if A1 != B1:
                        fusion.add_edge([i0, i1])
                        break

        fusion = fusion.transitive_closure()
        resu = []
        for s in fusion.connected_components():
            subg = [x for x in prod_dg if all(x[i] == v0[i] for i in factors_range
                                              if i not in s)]
            resu.append(Poset(prod_dg.subgraph(subg)))

        return resu

    def disjoint_union(self, other, labels='pairs'):
        """
        Return a poset isomorphic to disjoint union (also called direct
        sum) of the poset with ``other``.

        The disjoint union of `P` and `Q` is a poset that contains
        every element and relation from both `P` and `Q`, and where
        every element of `P` is incomparable to every element of `Q`.

        Mathematically, it is only defined when `P` and `Q` have no
        common element; here we force that by giving them different
        names in the resulting poset.

        INPUT:

        - ``other``, a poset.

        - ``labels`` - (defaults to 'pairs') If set to 'pairs', each
          element ``v`` in this poset will be named ``(0,v)`` and each
          element ``u`` in ``other`` will be named ``(1,u)`` in the
          result. If set to 'integers', the elements of the result
          will be relabeled with consecutive integers.

        EXAMPLES::

            sage: P1 = Poset({'a': 'b'})
            sage: P2 = Poset({'c': 'd'})
            sage: P = P1.disjoint_union(P2); P
            Finite poset containing 4 elements
            sage: sorted(P.cover_relations())
            [[(0, 'a'), (0, 'b')], [(1, 'c'), (1, 'd')]]
            sage: P = P1.disjoint_union(P2, labels='integers')
            sage: P.cover_relations()
            [[2, 3], [0, 1]]

            sage: N5 = posets.PentagonPoset(); N5
            Finite lattice containing 5 elements
            sage: N5.disjoint_union(N5)  # Union of lattices is not a lattice
            Finite poset containing 10 elements

        We show how to get literally direct sum with elements untouched::

            sage: P = P1.disjoint_union(P2).relabel(lambda x: x[1])
            sage: sorted(P.cover_relations())
            [['a', 'b'], ['c', 'd']]

        .. SEEALSO:: :meth:`connected_components`

        TESTS::

            sage: N5 = posets.PentagonPoset()
            sage: P0 = Poset()
            sage: N5.disjoint_union(P0).is_isomorphic(N5)
            True
            sage: P0.disjoint_union(P0)
            Finite poset containing 0 elements

            sage: A3 = posets.AntichainPoset(3)
            sage: A4 = posets.AntichainPoset(4)
            sage: A7 = posets.AntichainPoset(7)
            sage: A3.disjoint_union(A4).is_isomorphic(A7)
            True
        """
        if not hasattr(other, 'hasse_diagram'):
            raise TypeError("'other' is not a finite poset")
        return Poset(self.hasse_diagram().disjoint_union(other.hasse_diagram(),
                                                         labels=labels))

    def ordinal_product(self, other, labels='pairs'):
        r"""
        Return the ordinal product of ``self`` and ``other``.

        The ordinal product of two posets `P` and `Q` is a partial
        order on the Cartesian product of the underlying sets of `P`
        and `Q`, defined as follows (see [EnumComb1]_, p. 284).

        In the ordinal product, `(p,q) \leq (p',q')` if either `p \leq
        p'` or `p = p'` and `q \leq q'`.

        This construction is not symmetric in `P` and `Q`. Informally
        said we put a copy of `Q` in place of every element of `P`.

        INPUT:

        - ``other`` -- a poset

        - ``labels`` -- either ``'integers'`` or ``'pairs'`` (default); how
          the resulting poset will be labeled

        EXAMPLES::

            sage: P1 = Poset((['a', 'b'], [['a', 'b']]))
            sage: P2 = Poset((['c', 'd'], [['c', 'd']]))
            sage: P = P1.ordinal_product(P2); P
            Finite poset containing 4 elements
            sage: sorted(P.cover_relations())
            [[('a', 'c'), ('a', 'd')], [('a', 'd'), ('b', 'c')],
            [('b', 'c'), ('b', 'd')]]

        .. SEEALSO::

            :meth:`product`, :meth:`ordinal_sum`

        TESTS::

            sage: P1.ordinal_product(24)
            Traceback (most recent call last):
            ...
            TypeError: 'other' is not a finite poset
            sage: P1.ordinal_product(P2, labels='camembert')
            Traceback (most recent call last):
            ...
            ValueError: labels must be either 'pairs' or 'integers'

            sage: N5 = posets.PentagonPoset()
            sage: P0 = Poset()
            sage: N5.ordinal_product(P0) == P0
            True
            sage: P0.ordinal_product(N5) == P0
            True
            sage: P0.ordinal_product(P0) == P0
            True

            sage: A3 = posets.AntichainPoset(3)
            sage: A4 = posets.AntichainPoset(4)
            sage: A12 = posets.AntichainPoset(12)
            sage: A3.ordinal_product(A4).is_isomorphic(A12)
            True

            sage: C3 = posets.ChainPoset(3)
            sage: C4 = posets.ChainPoset(4)
            sage: C12 = posets.ChainPoset(12)
            sage: C3.ordinal_product(C4).is_isomorphic(C12)
            True
        """
        from sage.combinat.posets.lattices import LatticePoset, \
            FiniteLatticePoset

        if not hasattr(other, 'hasse_diagram'):
            raise TypeError("'other' is not a finite poset")
        othermax = other.maximal_elements()
        othermin = other.minimal_elements()

        dg = DiGraph()
        dg.add_vertices([(s, t) for s in self for t in other])
        dg.add_edges([((s, t), (s2, t2))
                      for s, s2 in self.cover_relations_iterator()
                      for t in othermax for t2 in othermin])
        dg.add_edges([((s, t), (s, t2))
                      for s in self
                      for t, t2 in other.cover_relations_iterator()])
        if labels == 'integers':
            dg.relabel()
        elif labels != 'pairs':
            raise ValueError("labels must be either 'pairs' or 'integers'")

        if (isinstance(self, FiniteLatticePoset) and
                isinstance(other, FiniteLatticePoset)):
            return LatticePoset(dg)
        return Poset(dg)

    def ordinal_sum(self, other, labels='pairs'):
        """
        Return a poset or (semi)lattice isomorphic to ordinal sum of the
        poset with ``other``.

        The ordinal sum of `P` and `Q` is a poset that contains every
        element and relation from both `P` and `Q`, and where every
        element of `P` is smaller than any element of `Q`.

        Mathematically, it is only defined when `P` and `Q` have no
        common element; here we force that by giving them different
        names in the resulting poset.

        The ordinal sum on lattices is a lattice; resp. for meet- and
        join-semilattices.

        INPUT:

        - ``other``, a poset.

        - ``labels`` - (defaults to 'pairs') If set to 'pairs', each
          element ``v`` in this poset will be named ``(0,v)`` and each
          element ``u`` in ``other`` will be named ``(1,u)`` in the
          result. If set to 'integers', the elements of the result
          will be relabeled with consecutive integers.

        EXAMPLES::

            sage: P1 = Poset( ([1, 2, 3, 4], [[1, 2], [1, 3], [1, 4]]) )
            sage: P2 = Poset( ([1, 2, 3,], [[2,1], [3,1]]) )
            sage: P3 = P1.ordinal_sum(P2); P3
            Finite poset containing 7 elements
            sage: len(P1.maximal_elements())*len(P2.minimal_elements())
            6
            sage: len(P1.cover_relations()+P2.cover_relations())
            5
            sage: len(P3.cover_relations()) # Every element of P2 is greater than elements of P1.
            11
            sage: P3.list()  # random
            [(0, 1), (0, 2), (0, 4), (0, 3), (1, 2), (1, 3), (1, 1)]
            sage: P4 = P1.ordinal_sum(P2, labels='integers')
            sage: P4.list()  # random
            [0, 1, 2, 3, 5, 6, 4]

        Return type depends on input types::

            sage: P = Poset({1:[2]}); P
            Finite poset containing 2 elements
            sage: JL = JoinSemilattice({1:[2]}); JL
            Finite join-semilattice containing 2 elements
            sage: L = LatticePoset({1:[2]}); L
            Finite lattice containing 2 elements
            sage: P.ordinal_sum(L)
            Finite poset containing 4 elements
            sage: L.ordinal_sum(JL)
            Finite join-semilattice containing 4 elements
            sage: L.ordinal_sum(L)
            Finite lattice containing 4 elements

        .. SEEALSO::

            :meth:`ordinal_summands`, :meth:`disjoint_union`,
            :meth:`sage.combinat.posets.lattices.FiniteLatticePoset.vertical_composition`

        TESTS::

            sage: N5 = posets.PentagonPoset()
            sage: P0 = LatticePoset({})
            sage: N5.ordinal_sum(P0).is_isomorphic(N5)
            True
            sage: P0.ordinal_sum(P0)
            Finite lattice containing 0 elements
        """
        from sage.combinat.posets.lattices import LatticePoset, \
            JoinSemilattice, MeetSemilattice, FiniteLatticePoset, \
            FiniteMeetSemilattice, FiniteJoinSemilattice

        if not hasattr(other, 'hasse_diagram'):
            raise TypeError("'other' is not a finite poset")
        G = self.hasse_diagram().disjoint_union(other.hasse_diagram())
        selfmax = self.maximal_elements()
        othermin = other.minimal_elements()
        for u in selfmax:
            for v in othermin:
                G.add_edge((0, u), (1, v))
        if labels == 'integers':
            G.relabel()
        elif labels != 'pairs':
            raise ValueError("labels must be either 'pairs' or 'integers'")

        if (isinstance(self, FiniteLatticePoset) and
                isinstance(other, FiniteLatticePoset)):
            return LatticePoset(G)
        if (isinstance(self, FiniteMeetSemilattice) and
                isinstance(other, FiniteMeetSemilattice)):
            return MeetSemilattice(G)
        if (isinstance(self, FiniteJoinSemilattice) and
                isinstance(other, FiniteJoinSemilattice)):
            return JoinSemilattice(G)
        return Poset(G)

    def star_product(self, other, labels='pairs'):
        r"""
        Return a poset isomorphic to the star product of the
        poset with ``other``.

        Both this poset and ``other`` are expected to be bounded
        and have at least two elements.

        Let `P` be a poset with top element `\top_P` and `Q` be a poset
        with bottom element `\bot_Q`. The star product of
        `P` and `Q` is the ordinal sum of `P \setminus \top_P` and
        `Q \setminus \bot_Q`.

        Mathematically, it is only defined when `P` and `Q` have no
        common elements; here we force that by giving them different
        names in the resulting poset.

        INPUT:

        - ``other`` -- a poset.

        - ``labels`` -- (defaults to 'pairs') If set to 'pairs', each
          element ``v`` in this poset will be named ``(0, v)`` and each
          element ``u`` in ``other`` will be named ``(1, u)`` in the
          result. If set to 'integers', the elements of the result
          will be relabeled with consecutive integers.

        EXAMPLES:

        This is mostly used to combine two Eulerian posets to third one,
        and makes sense for graded posets only::

            sage: B2 = posets.BooleanLattice(2)
            sage: B3 = posets.BooleanLattice(3)
            sage: P = B2.star_product(B3); P
            Finite poset containing 10 elements
            sage: P.is_eulerian()
            True

        We can get elements as pairs or as integers::

            sage: ABC = Poset({'a': ['b'], 'b': ['c']})
            sage: XYZ = Poset({'x': ['y'], 'y': ['z']})
            sage: ABC.star_product(XYZ).list()
            [(0, 'a'), (0, 'b'), (1, 'y'), (1, 'z')]
            sage: sorted(ABC.star_product(XYZ, labels='integers'))
            [0, 1, 2, 3]

        TESTS::

            sage: C0 = Poset()
            sage: C1 = Poset({0: []})
            sage: C2 = Poset({0: [1]})
            sage: C2.star_product(42)
            Traceback (most recent call last):
            ...
            TypeError: 'other' is not a finite poset
            sage: C2.star_product(C0)
            Traceback (most recent call last):
            ...
            ValueError: 'other' is not bounded
            sage: C0.star_product(C2)
            Traceback (most recent call last):
            ...
            ValueError: the poset is not bounded
            sage: C2.star_product(C1)
            Traceback (most recent call last):
            ...
            ValueError: 'other' has less than two elements
            sage: C1.star_product(C2)
            Traceback (most recent call last):
            ...
            ValueError: the poset has less than two elements
        """
        if not hasattr(other, 'hasse_diagram'):
            raise TypeError("'other' is not a finite poset")
        if not self.is_bounded():
            raise ValueError("the poset is not bounded")
        if not other.is_bounded():
            raise ValueError("'other' is not bounded")
        if self.cardinality() < 2:
            raise ValueError("the poset has less than two elements")
        if other.cardinality() < 2:
            raise ValueError("'other' has less than two elements")
        if labels not in ['pairs', 'integers']:
            raise ValueError("labels must be either 'pairs' or 'integers'")

        G = self.hasse_diagram().disjoint_union(other.hasse_diagram())
        selfmax = self.lower_covers(self.top())
        othermin = other.upper_covers(other.bottom())
        G.delete_vertex((0, self.top()))
        G.delete_vertex((1, other.bottom()))
        for u in selfmax:
            for v in othermin:
                G.add_edge((0, u), (1, v))
        if labels == 'integers':
            G.relabel()
        return Poset(G)

    def lexicographic_sum(self, P):
        r"""
        Return the lexicographic sum using this poset as index.

        In the lexicographic sum of posets `P_t` by index poset `T`
        we have `x \le y` if either `x \le y` in `P_t`
        for some `t \in T`, or `x \in P_i`, `y \in P_j` and
        `i \le j` in `T`.

        Informally said we substitute every element of `T` by corresponding
        poset `P_t`.

        Mathematically, it is only defined when all `P_t` have no
        common element; here we force that by giving them different
        names in the resulting poset.

        :meth:`disjoint_union` and :meth:`ordinal_sum` are special cases
        of lexicographic sum where the index poset is an (anti)chain.
        :meth:`ordinal_product` is a special case where every `P_t` is same
        poset.

        INPUT:

        - ``P`` -- dictionary whose keys are elements of this poset, values are posets

        EXAMPLES::

            sage: N = Poset({1: [3, 4], 2: [4]})
            sage: P = {1: posets.PentagonPoset(), 2: N, 3: posets.ChainPoset(3), 4: posets.AntichainPoset(4)}
            sage: NP = N.lexicographic_sum(P); NP
            Finite poset containing 16 elements
            sage: sorted(NP.minimal_elements())
            [(1, 0), (2, 1), (2, 2)]

        TESTS::

            sage: from collections import defaultdict
            sage: P = defaultdict(lambda: Poset({3: [4]}))
            sage: P[1] = Poset({5: [6]})
            sage: T = Poset({1: [2]})
            sage: T.lexicographic_sum(P)
            Finite poset containing 4 elements

            sage: P[1] = Poset()
            sage: T.lexicographic_sum(P)
            Traceback (most recent call last):
            ...
            ValueError: lexicographic sum of the empty poset is not defined

            sage: P = {1: T, 2: T, 3: T}
            sage: T.lexicographic_sum(P)
            Traceback (most recent call last):
            ...
            ValueError: keys of dict P does not match to elements of the poset
            sage: P = {1: T}
            sage: T.lexicographic_sum(P)
            Traceback (most recent call last):
            ...
            ValueError: keys of dict P does not match to elements of the poset

            sage: P = {1: Poset({1: []}, facade=True), 2: Poset({1: []}, facade=False)}
            sage: T.lexicographic_sum(P)
            Traceback (most recent call last):
            ...
            TypeError: mixing facade and non-facade posets is not defined

            sage: T = Poset({42: []}, facade=False)
            sage: P = {T[0]: Poset({57: []}, facade=False)}
            sage: T.lexicographic_sum(P)._is_facade
            False
            sage: T = Poset({42: []}, facade=True)
            sage: P = {T[0]: Poset({57: []}, facade=True)}
            sage: T.lexicographic_sum(P)._is_facade
            True
        """
        # P might be defaultdict, hence the test
        if type(P) == type({}):
            if set(P) != set(self):
                raise ValueError("keys of dict P does not match to elements of the poset")

        d = DiGraph()
        for t in self:
            if P[t]._is_facade != self._is_facade:
                raise TypeError("mixing facade and non-facade posets is not defined")
            if P[t].cardinality() == 0:
                raise ValueError("lexicographic sum of the empty poset is not defined")
            part = P[t].hasse_diagram()
            part.relabel(lambda x: (t, x), inplace=True)
            d = d.union(part)
        for a, b in self.cover_relations_iterator():
            for l in P[a].maximal_elements():
                for u in P[b].minimal_elements():
                    d.add_edge((a, l), (b, u))
        return Poset(d, facade=self._is_facade)

    def dual(self):
        r"""
        Return the dual poset of the given poset.

        In the dual of a poset `P` we have `x \le y` iff `y \le x` in `P`.

        EXAMPLES::

            sage: P = Poset({1: [2, 3], 3: [4]})
            sage: P.cover_relations()
            [[1, 2], [1, 3], [3, 4]]
            sage: Q = P.dual()
            sage: Q.cover_relations()
            [[4, 3], [3, 1], [2, 1]]

        Dual of a lattice is a lattice; dual of a meet-semilattice is
        join-semilattice and vice versa. Also the dual of a (non-)facade poset
        is again (non-)facade::

            sage: V = MeetSemilattice({1: [2, 3]}, facade=False)
            sage: A = V.dual(); A
            Finite join-semilattice containing 3 elements
            sage: A(2) < A(1)
            True

        .. SEEALSO:: :meth:`~sage.categories.finite_posets.FinitePosets.ParentMethods.is_self_dual`

        TESTS::

            sage: Poset().dual() == Poset()  # Test the empty poset
            True
        """
        if self._with_linear_extension:
            elements = reversed(self._elements)
        else:
            elements = None
        H = self._hasse_diagram.relabel({i: x for i, x in enumerate(self._elements)},
                                        inplace=False)
        return self._dual_class(H.reverse(),
                                elements=elements,
                                category=self.category(),
                                facade=self._is_facade)

    def with_bounds(self, labels=('bottom', 'top')):
        r"""
        Return the poset with bottom and top elements adjoined.

        This function adds top and bottom elements to the poset.
        It will always add elements, it does not check if the poset
        already has a bottom or a top element.

        For lattices and semilattices this function returns a lattice.

        INPUT:

        - ``labels`` -- A pair of elements to use as a bottom and top
          element of the poset. Default is strings ``'bottom'`` and
          ``'top'``. Either of them can be ``None``, and then a new
          bottom or top element will not be added.

        EXAMPLES::

            sage: V = Poset({0: [1, 2]})
            sage: trafficsign = V.with_bounds(); trafficsign
            Finite poset containing 5 elements
            sage: trafficsign.list()
            ['bottom', 0, 1, 2, 'top']
            sage: trafficsign = V.with_bounds(labels=(-1, -2))
            sage: trafficsign.cover_relations()
            [[-1, 0], [0, 1], [0, 2], [1, -2], [2, -2]]

            sage: Y = V.with_bounds(labels=(-1, None))
            sage: Y.cover_relations()
            [[-1, 0], [0, 1], [0, 2]]

            sage: P = posets.PentagonPoset()  # A lattice
            sage: P.with_bounds()
            Finite lattice containing 7 elements

            sage: P = posets.PentagonPoset(facade=False)
            sage: P.with_bounds()
            Finite lattice containing 7 elements

        .. SEEALSO::

            :meth:`without_bounds` for the reverse operation

        TESTS::

            sage: P = Poset().with_bounds()
            sage: P.cover_relations()
            [['bottom', 'top']]

            sage: L = LatticePoset({}).with_bounds(); L
            Finite lattice containing 2 elements
            sage: L.meet_irreducibles()  # Trac 21543
            ['bottom']

            sage: Poset().with_bounds((None, 1))
            Finite poset containing 1 elements
            sage: LatticePoset().with_bounds((None, 1))
            Finite lattice containing 1 elements
            sage: MeetSemilattice().with_bounds((None, 1))
            Finite lattice containing 1 elements
            sage: JoinSemilattice().with_bounds((None, 1))
            Finite join-semilattice containing 1 elements
            sage: Poset().with_bounds((1, None))
            Finite poset containing 1 elements
            sage: LatticePoset().with_bounds((1, None))
            Finite lattice containing 1 elements
            sage: MeetSemilattice().with_bounds((1, None))
            Finite meet-semilattice containing 1 elements
            sage: JoinSemilattice().with_bounds((1, None))
            Finite lattice containing 1 elements

            sage: P = Poset({0: []})
            sage: L = LatticePoset({0: []})
            sage: ML = MeetSemilattice({0: []})
            sage: JL = JoinSemilattice({0: []})
            sage: P.with_bounds((None, None))
            Finite poset containing 1 elements
            sage: L.with_bounds((None, None))
            Finite lattice containing 1 elements
            sage: ML.with_bounds((None, None))
            Finite meet-semilattice containing 1 elements
            sage: JL.with_bounds((None, None))
            Finite join-semilattice containing 1 elements
            sage: P.with_bounds((1, None))
            Finite poset containing 2 elements
            sage: L.with_bounds((1, None))
            Finite lattice containing 2 elements
            sage: ML.with_bounds((1, None))
            Finite meet-semilattice containing 2 elements
            sage: JL.with_bounds((1, None))
            Finite lattice containing 2 elements
            sage: P.with_bounds((None, 1))
            Finite poset containing 2 elements
            sage: L.with_bounds((None, 1))
            Finite lattice containing 2 elements
            sage: ML.with_bounds((None, 1))
            Finite lattice containing 2 elements
            sage: JL.with_bounds((None, 1))
            Finite join-semilattice containing 2 elements

            sage: posets.PentagonPoset().with_bounds(labels=(4, 5))
            Traceback (most recent call last):
            ...
            ValueError: the poset already has element 4
        """
        if len(labels) != 2:
            raise TypeError("labels must be a pair")
        new_min, new_max = labels
        if new_min in self:
            raise ValueError("the poset already has element %s" % new_min)
        if new_max in self:
            raise ValueError("the poset already has element %s" % new_max)

        from sage.combinat.posets.lattices import LatticePoset, \
            JoinSemilattice, MeetSemilattice, FiniteLatticePoset, \
            FiniteMeetSemilattice, FiniteJoinSemilattice
        if (isinstance(self, FiniteLatticePoset) or
            (isinstance(self, FiniteMeetSemilattice) and new_max is not None) or
            (isinstance(self, FiniteJoinSemilattice) and new_min is not None)):
            constructor = LatticePoset
        elif isinstance(self, FiniteMeetSemilattice):
            constructor = MeetSemilattice
        elif isinstance(self, FiniteJoinSemilattice):
            constructor = JoinSemilattice
        else:
            constructor = Poset

        if self.cardinality() == 0:
            if new_min is None and new_max is None:
                return constructor(facade=self._is_facade)
            if new_min is None:
                return constructor({new_min: []}, facade=self._is_facade)
            if new_max is None:
                return constructor({new_max: []}, facade=self._is_facade)
            return constructor({new_min: [new_max]}, facade=self._is_facade)
        if self._is_facade:
            D = self.hasse_diagram()
        else:
            # we recreate the poset, so that the elements no longer
            # reference the original poset.
            D = Poset(self, facade=True).hasse_diagram()

        if new_min is not None:
            D.add_edges([(new_min, e) for e in D.sources()])
        if new_max is not None:
            D.add_edges([(e, new_max) for e in D.sinks()])
        return constructor(D, facade=self._is_facade)

    def without_bounds(self):
        """
        Return the poset without its top and bottom elements.

        This is useful as an input for the method :meth:`order_complex`.

        If there is either no top or no bottom element, this
        raises a ``TypeError``.

        EXAMPLES::

            sage: P = posets.PentagonPoset()
            sage: Q = P.without_bounds(); Q
            Finite poset containing 3 elements
            sage: Q.cover_relations()
            [[2, 3]]

            sage: P = posets.DiamondPoset(5)
            sage: Q = P.without_bounds(); Q
            Finite poset containing 3 elements
            sage: Q.cover_relations()
            []

        .. SEEALSO::

            :meth:`with_bounds` for the reverse operation

        TESTS::

            sage: P = Poset({1:[2],3:[2,4]})
            sage: P.without_bounds()
            Traceback (most recent call last):
            ...
            TypeError: the poset is missing either top or bottom

            sage: P = Poset({1:[]})
            sage: P.without_bounds()
            Finite poset containing 0 elements

            sage: P = Poset({})
            sage: P.without_bounds()
            Traceback (most recent call last):
            ...
            TypeError: the poset is missing either top or bottom
        """
        if self.is_bounded():
            g = copy(self._hasse_diagram)
            g.delete_vertices([0, g.order() - 1])
            g.relabel(self._vertex_to_element)
            return Poset(g, facade=self._is_facade)
        raise TypeError('the poset is missing either top or bottom')

    def relabel(self, relabeling=None):
        r"""
        Return a copy of this poset with its elements relabeled.

        INPUT:

        - ``relabeling`` -- a function, dictionary, list or tuple

        The given function or dictionary must map each (non-wrapped)
        element of ``self`` to some distinct object. The given list or tuple
        must be made of distinct objects.

        When the input is a list or a tuple, the relabeling uses
        the total ordering of the elements of the poset given by
        ``list(self)``.

        If no relabeling is given, the poset is relabeled by integers
        from `0` to `n-1` according to one of its linear extensions. This means
        that `i<j` as integers whenever `i<j` in the relabeled poset.

        EXAMPLES:

        Relabeling using a function::

            sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
            sage: P.list()
            [1, 2, 3, 4, 6, 12]
            sage: P.cover_relations()
            [[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
            sage: Q = P.relabel(lambda x: x+1)
            sage: Q.list()
            [2, 3, 4, 5, 7, 13]
            sage: Q.cover_relations()
            [[2, 3], [2, 4], [3, 5], [3, 7], [4, 7], [5, 13], [7, 13]]

        Relabeling using a dictionary::

            sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True, facade=False)
            sage: relabeling = {c.element:i for (i,c) in enumerate(P)}
            sage: relabeling
            {1: 0, 2: 1, 3: 2, 4: 3, 6: 4, 12: 5}
            sage: Q = P.relabel(relabeling)
            sage: Q.list()
            [0, 1, 2, 3, 4, 5]
            sage: Q.cover_relations()
            [[0, 1], [0, 2], [1, 3], [1, 4], [2, 4], [3, 5], [4, 5]]

        Mind the ``c.element``; this is because the relabeling is
        applied to the elements of the poset without the wrapping.
        Thanks to this convention, the same relabeling function can
        be used both for facade or non facade posets.

        Relabeling using a list::

            sage: P = posets.PentagonPoset()
            sage: list(P)
            [0, 1, 2, 3, 4]
            sage: P.cover_relations()
            [[0, 1], [0, 2], [1, 4], [2, 3], [3, 4]]
            sage: Q = P.relabel(list('abcde'))
            sage: Q.cover_relations()
            [['a', 'b'], ['a', 'c'], ['b', 'e'], ['c', 'd'], ['d', 'e']]

        Default behaviour is increasing relabeling::

            sage: a2 = posets.ChainPoset(2)
            sage: P = a2 * a2
            sage: Q = P.relabel()
            sage: Q.cover_relations()
            [[0, 1], [0, 2], [1, 3], [2, 3]]

        Relabeling a (semi)lattice gives a (semi)lattice::

            sage: P = JoinSemilattice({0: [1]})
            sage: P.relabel(lambda n: n+1)
            Finite join-semilattice containing 2 elements

        .. NOTE::

            As can be seen in the above examples, the default linear
            extension of ``Q`` is that of ``P`` after relabeling. In
            particular, ``P`` and ``Q`` share the same internal Hasse
            diagram.

        TESTS:

        Test non-facade poset::

            sage: P = Poset({3: [2]}, facade=False)
            sage: Q = P.relabel(lambda x: chr(ord('a')+x))
            sage: Q('c') < Q('d')
            False

        The following checks that :trac:`14019` has been fixed::

            sage: d = DiGraph({2:[1],3:[1]})
            sage: p1 = Poset(d)
            sage: p2 = p1.relabel({1:1,2:3,3:2})
            sage: p1.hasse_diagram() == p2.hasse_diagram()
            True
            sage: p1 == p2
            True

            sage: d = DiGraph({2:[1],3:[1]})
            sage: p1 = Poset(d)
            sage: p2 = p1.relabel({1:2,2:3,3:1})
            sage: p3 = p2.relabel({2:1,1:2,3:3})
            sage: p1.hasse_diagram() == p3.hasse_diagram()
            True
            sage: p1 == p3
            True
        """
        from sage.combinat.posets.lattices import (FiniteLatticePoset,
             FiniteMeetSemilattice, FiniteJoinSemilattice)

        if isinstance(self, FiniteLatticePoset):
            constructor = FiniteLatticePoset
        elif isinstance(self, FiniteMeetSemilattice):
            constructor = FiniteMeetSemilattice
        elif isinstance(self, FiniteJoinSemilattice):
            constructor = FiniteJoinSemilattice
        else:
            constructor = FinitePoset

        if relabeling is None:
            return constructor(self._hasse_diagram, category=self.category(),
                               facade=self._is_facade)

        if isinstance(relabeling, (list, tuple)):
            relabeling = {i: relabeling[i]
                          for i in range(len(self._elements))}
        else:
            if isinstance(relabeling, dict):
                relabeling = relabeling.__getitem__
            relabeling = {i: relabeling(x)
                          for i, x in enumerate(self._elements)}

        if not self._with_linear_extension:
            elements = None
        else:
            elements = tuple(relabeling[self._element_to_vertex(x)]
                             for x in self._elements)

        return constructor(self._hasse_diagram.relabel(relabeling,
                                                       inplace=False),
                           elements=elements, category=self.category(),
                           facade=self._is_facade)

    def canonical_label(self, algorithm=None):
        r"""
        Return the unique poset on the labels `\{0, \ldots, n-1\}` (where `n`
        is the number of elements in the poset) that is isomorphic to this
        poset and invariant in the isomorphism class.

        INPUT:

        - ``algorithm`` -- string (optional); a parameter forwarded
          to underlying graph function to select the algorithm to use

        EXAMPLES::

            sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
            sage: P.list()
            [1, 2, 3, 4, 6, 12]
            sage: Q = P.canonical_label()
            sage: sorted(Q.list())
            [0, 1, 2, 3, 4, 5]
            sage: Q.is_isomorphic(P)
            True

        Canonical labeling of (semi)lattice returns (semi)lattice::

            sage: D = DiGraph({'a':['b','c']})
            sage: P = Poset(D)
            sage: ML = MeetSemilattice(D)
            sage: P.canonical_label()
            Finite poset containing 3 elements
            sage: ML.canonical_label()
            Finite meet-semilattice containing 3 elements

        .. SEEALSO::

            - Canonical labeling of directed graphs:
              :meth:`~sage.graphs.generic_graph.GenericGraph.canonical_label()`

        TESTS::

            sage: P = Poset(digraphs.Path(10), linear_extension=True)
            sage: Q = P.canonical_label()
            sage: Q.linear_extension() # random
            [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
            sage: Q.cover_relations() # random
            [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]]
            sage: P = Poset(digraphs.Path(10))
            sage: Q = P.canonical_label()
            sage: Q.linear_extension() # random
            [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
            sage: Q.is_isomorphic(P)
            True

            sage: Poset().canonical_label()  # Test the empty poset
            Finite poset containing 0 elements

            sage: D2 = posets.DiamondPoset(4).canonical_label(algorithm='bliss')  # optional: bliss
            sage: B2 = posets.BooleanLattice(2).canonical_label(algorithm='bliss')  # optional: bliss
            sage: D2 == B2  # optional: bliss
            True
        """
        canonical_label = self._hasse_diagram.canonical_label(certificate=True,
                                                              algorithm=algorithm)[1]
        canonical_label = {self._elements[v]: i for v, i in canonical_label.items()}
        return self.relabel(canonical_label)

    def with_linear_extension(self, linear_extension):
        """
        Return a copy of ``self`` with a different default linear extension.

        EXAMPLES::

            sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
            sage: P.cover_relations()
            [[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
            sage: list(P)
            [1, 2, 3, 4, 6, 12]
            sage: Q = P.with_linear_extension([1,3,2,6,4,12])
            sage: list(Q)
            [1, 3, 2, 6, 4, 12]
            sage: Q.cover_relations()
            [[1, 3], [1, 2], [3, 6], [2, 6], [2, 4], [6, 12], [4, 12]]

        TESTS:

        We check that we can pass in a list of elements of ``P`` instead::

            sage: Q = P.with_linear_extension([P(_) for _ in [1,3,2,6,4,12]])
            sage: list(Q)
            [1, 3, 2, 6, 4, 12]
            sage: Q.cover_relations()
            [[1, 3], [1, 2], [3, 6], [2, 6], [2, 4], [6, 12], [4, 12]]

        We check that this works for facade posets too::

            sage: P = Poset((divisors(12), attrcall("divides")), facade=True)
            sage: Q = P.with_linear_extension([1,3,2,6,4,12]); Q
            Finite poset containing 6 elements with distinguished linear extension
            sage: list(Q)
            [1, 3, 2, 6, 4, 12]
            sage: Q.cover_relations()
            [[1, 3], [1, 2], [3, 6], [2, 6], [2, 4], [6, 12], [4, 12]]
            sage: sorted(Q.cover_relations()) == sorted(P.cover_relations())
            True

        (Semi)lattice remains (semi)lattice with new linear extension::

            sage: L = LatticePoset(P)
            sage: Q = L.with_linear_extension([1,3,2,6,4,12]); Q
            Finite lattice containing 6 elements with distinguished linear extension

        .. NOTE::

            With the current implementation, this requires relabeling
            the internal :class:`DiGraph` which is `O(n+m)`, where `n`
            is the number of elements and `m` the number of cover relations.
        """
        new_vertices = [self._element_to_vertex(element) for element in linear_extension]
        vertex_relabeling = dict(zip(new_vertices, linear_extension))
        # Hack to get the actual class, not the categorified class
        constructor = self.__class__.__base__
        return constructor(self._hasse_diagram.relabel(vertex_relabeling, inplace=False),
                           elements=linear_extension,
                           category=self.category(),
                           facade=self._is_facade)

    def graphviz_string(self, graph_string="graph", edge_string="--"):
        r"""
        Return a representation in the DOT language, ready to render in
        graphviz.

        See http://www.graphviz.org/doc/info/lang.html for more information
        about graphviz.

        EXAMPLES::

            sage: P = Poset({'a':['b'],'b':['d'],'c':['d'],'d':['f'],'e':['f'],'f':[]})
            sage: print(P.graphviz_string())
            graph {
            "f";"d";"b";"a";"c";"e";
            "f"--"e";"d"--"c";"b"--"a";"d"--"b";"f"--"d";
            }
        """
        s = '%s {\n' % graph_string
        for v in reversed(self.list()):
            s += '"%s";' % v
        s += '\n'
        for u, v in self.cover_relations_iterator():
            s += '"%s"%s"%s";' % (v, edge_string, u)
        s += "\n}"
        return s

    def subposet(self, elements):
        """
        Return the poset containing given elements with partial order
        induced by this poset.

        EXAMPLES::

            sage: P = Poset({'a': ['c', 'd'], 'b': ['d','e'], 'c': ['f'],
            ....:            'd': ['f'], 'e': ['f']})
            sage: Q = P.subposet(['a', 'b', 'f']); Q
            Finite poset containing 3 elements
            sage: Q.cover_relations()
            [['b', 'f'], ['a', 'f']]

        A subposet of a non-facade poset is again a non-facade poset::

            sage: P = posets.PentagonPoset(facade=False)
            sage: Q = P.subposet([0, 1, 2, 4])
            sage: Q(1) < Q(2)
            False

        TESTS::

            sage: P = Poset({'a': ['b'], 'b': ['c']})
            sage: P.subposet(('a', 'b', 'c'))
            Finite poset containing 3 elements
            sage: P.subposet([])
            Finite poset containing 0 elements
            sage: P.subposet(["a","b","x"])
            Traceback (most recent call last):
            ...
            ValueError: element (=x) not in poset
            sage: P.subposet(3)
            Traceback (most recent call last):
            ...
            TypeError: 'sage.rings.integer.Integer' object is not iterable
        """
        H = self._hasse_diagram
        elms = sorted(set(self._element_to_vertex(e) for e in elements))

        if not elms:
            return Poset()

        relations = []
        lt = [set() for _ in range(elms[-1] + 1)]

        for i in range(elms[0], elms[-1] + 1):
            for low in H.neighbor_in_iterator(i):
                lt[i].update(lt[low])
            if i in elms:
                relations += [(x, i) for x in lt[i]]
                lt[i] = set([i])

        g = DiGraph([elms, relations], format='vertices_and_edges')
        if self._is_facade:
            g.relabel(self._vertex_to_element, inplace=True)
        else:
            g.relabel(lambda v: self._vertex_to_element(v).element, inplace=True)
        return Poset(g, cover_relations=False, facade=self._is_facade)

    def random_subposet(self, p):
        """
        Return a random subposet that contains each element with
        probability ``p``.

        EXAMPLES::

            sage: P = posets.BooleanLattice(3)
            sage: set_random_seed(0)  # Results are reproducible
            sage: Q = P.random_subposet(0.5)
            sage: Q.cover_relations()
            [[0, 2], [0, 5], [2, 3], [3, 7], [5, 7]]

        TESTS::

            sage: P = posets.IntegerPartitions(4)
            sage: P.random_subposet(1) == P
            True
        """
        from sage.misc.randstate import current_randstate
        random = current_randstate().python_random().random
        elements = []
        p = float(p)
        if p < 0 or p > 1:
            raise ValueError("probability p must be in [0..1]")
        for v in self:
            if random() <= p:
                elements.append(v)
        return self.subposet(elements)

    def random_order_ideal(self, direction='down'):
        """
        Return a random order ideal with uniform probability.

        INPUT:

        - ``direction`` -- ``'up'``, ``'down'`` or ``'antichain'``
          (default: ``'down'``)

        OUTPUT:

        A randomly selected order ideal (or order filter if
        ``direction='up'``, or antichain if ``direction='antichain'``)
        where all order ideals have equal probability of occurring.

        ALGORITHM:

        Uses the coupling from the past algorithm described in [Propp1997]_.

        EXAMPLES::

            sage: P = posets.BooleanLattice(3)
            sage: P.random_order_ideal()  # random
            [0, 1, 2, 3, 4, 5, 6]
            sage: P.random_order_ideal(direction='up')  # random
            [6, 7]
            sage: P.random_order_ideal(direction='antichain')  # random
            [1, 2]

            sage: P = posets.TamariLattice(5)
            sage: a = P.random_order_ideal('antichain')
            sage: P.is_antichain_of_poset(a)
            True
            sage: a = P.random_order_ideal('up')
            sage: P.is_order_filter(a)
            True
            sage: a = P.random_order_ideal('down')
            sage: P.is_order_ideal(a)
            True
        """
        from sage.misc.randstate import current_randstate
        from sage.misc.randstate import seed
        from sage.misc.randstate import random
        hd = self._hasse_diagram
        n = len(hd)
        lower_covers = [list(hd.lower_covers_iterator(i)) for i in range(n)]
        upper_covers = [list(hd.upper_covers_iterator(i)) for i in range(n)]
        count = n
        seedlist = [(current_randstate().long_seed(), count)]
        while True:
            # states are 0 -- in order ideal
            # 1 -- not in order ideal 2 -- undecided
            state = [2] * n
            for currseed, count in seedlist:
                with seed(currseed):
                    for _ in range(count):
                        for element in range(n):
                            if random() % 2 == 1:
                                s = [state[i] for i in lower_covers[element]]
                                if 1 not in s:
                                    if 2 not in s:
                                        state[element] = 0
                                    elif state[element] == 1:
                                        state[element] = 2
                            else:
                                s = [state[i] for i in upper_covers[element]]
                                if 0 not in s:
                                    if 2 not in s:
                                        state[element] = 1
                                    elif state[element] == 0:
                                        state[element] = 2
            if all(x != 2 for x in state):
                break
            count = seedlist[0][1] * 2
            seedlist.insert(0, (current_randstate().long_seed(), count))
        if direction == 'up':
            return [self._vertex_to_element(i) for i, x in enumerate(state) if x == 1]
        if direction == 'antichain':
            return [self._vertex_to_element(i) for i, x in enumerate(state)
                    if x == 0 and all(state[j] == 1 for j in hd.upper_covers_iterator(i))]
        if direction != 'down':
            raise ValueError("direction must be 'up', 'down' or 'antichain'")
        return [self._vertex_to_element(i) for i, x in enumerate(state) if x == 0]

    def random_maximal_chain(self):
        """
        Return a random maximal chain of the poset.

        The distribution is not uniform.

        EXAMPLES::

            sage: set_random_seed(0)  # results are reproduceable
            sage: P = posets.BooleanLattice(4)
            sage: P.random_maximal_chain()
            [0, 2, 10, 11, 15]

        TESTS::

            sage: Poset().random_maximal_chain()
            []
            sage: Poset({42: []}).random_maximal_chain()
            [42]
        """
        from sage.misc.prandom import randint

        if self.cardinality() == 0:
            return []

        mins = self.minimal_elements()
        new = mins[randint(0, len(mins) - 1)]
        result = [new]
        nexts = self.upper_covers(new)
        while nexts:
            new = nexts[randint(0, len(nexts) - 1)]
            result.append(new)
            nexts = self.upper_covers(new)
        return result

    def random_maximal_antichain(self):
        """
        Return a random maximal antichain of the poset.

        The distribution is not uniform.

        EXAMPLES::

            sage: set_random_seed(0)  # results are reproduceable
            sage: P = posets.BooleanLattice(4)
            sage: P.random_maximal_antichain()
            [1, 8, 2, 4]

        TESTS::

            sage: Poset().random_maximal_antichain()
            []
            sage: Poset({42: []}).random_maximal_antichain()
            [42]
        """
        H = self.hasse_diagram()
        result = []

        while H.order():
            new = H.random_vertex()
            result.append(new)
            down = list(H.depth_first_search(new, neighbors=H.neighbor_in_iterator))
            up = list(H.depth_first_search(new))
            H.delete_vertices(down+up)

        return result

    def random_linear_extension(self):
        """
        Return a random linear extension of the poset.

        The distribution is not uniform.

        EXAMPLES::

            sage: set_random_seed(0)  # results are reproduceable
            sage: P = posets.BooleanLattice(4)
            sage: P.random_linear_extension()
            [0, 2, 8, 1, 9, 4, 5, 10, 6, 12, 14, 13, 3, 7, 11, 15]

        TESTS::

            sage: Poset().random_linear_extension()
            []
            sage: Poset({42: []}).random_linear_extension()
            [42]
        """
        from sage.misc.prandom import randint

        H = self._hasse_diagram
        result = []
        indegs = list(H.in_degree_iterator())
        mins = H.sources()

        for _ in range(H.order()):
            new_index = randint(0, len(mins)-1)
            new = mins[new_index]
            result.append(new)
            mins = mins[:new_index]+mins[new_index+1:]
            for u in H.neighbor_out_iterator(new):
                indegs[u] -= 1
                if indegs[u] == 0:
                    mins.append(u)

        return [self._vertex_to_element(v) for v in result]

    def order_filter(self, elements):
        r"""
        Return the order filter generated by the elements of an
        iterable ``elements``.

        `I` is an order filter if, for any `x` in `I` and `y` such that
        `y \ge x`, then `y` is in `I`. This is also called upper set or
        upset.

        EXAMPLES::

            sage: P = Poset((divisors(1000), attrcall("divides")))
            sage: P.order_filter([20, 25])
            [20, 40, 25, 50, 100, 200, 125, 250, 500, 1000]

        .. SEEALSO::

            :meth:`order_ideal`, :meth:`~sage.categories.posets.Posets.ParentMethods.principal_order_filter`.

        TESTS::

            sage: P = Poset()  # Test empty poset
            sage: P.order_filter([])
            []
            sage: C = posets.ChainPoset(5)
            sage: C.order_filter([])
            []
        """
        vertices = sorted(map(self._element_to_vertex, elements))
        of = self._hasse_diagram.order_filter(vertices)
        return [self._vertex_to_element(_) for _ in of]

    def order_ideal(self, elements):
        r"""
        Return the order ideal generated by the elements of an
        iterable ``elements``.

        `I` is an order ideal if, for any `x` in `I` and `y` such that
        `y \le x`, then `y` is in `I`. This is also called lower set or
        downset.

        EXAMPLES::

            sage: P = Poset((divisors(1000), attrcall("divides")))
            sage: P.order_ideal([20, 25])
            [1, 2, 4, 5, 10, 20, 25]

        .. SEEALSO::

            :meth:`order_filter`, :meth:`~sage.categories.posets.Posets.ParentMethods.principal_order_ideal`.

        TESTS::

            sage: P = Poset()  # Test empty poset
            sage: P.order_ideal([])
            []
            sage: C = posets.ChainPoset(5)
            sage: C.order_ideal([])
            []
        """
        vertices = [self._element_to_vertex(_) for _ in elements]
        oi = self._hasse_diagram.order_ideal(vertices)
        return [self._vertex_to_element(_) for _ in oi]

    def order_ideal_cardinality(self, elements):
        r"""
        Return the cardinality of the order ideal generated by ``elements``.

        The elements `I` is an order ideal if, for any `x \in I` and `y`
        such that `y \le x`, then `y \in I`.

        EXAMPLES::

            sage: P = posets.BooleanLattice(4)
            sage: P.order_ideal_cardinality([7,10])
            10
        """
        vertices = [self._element_to_vertex(elmt) for elmt in elements]
        return self._hasse_diagram.order_ideal_cardinality(vertices)

    def order_ideal_plot(self, elements):
        r"""
        Return a plot of the order ideal generated by the elements of an
        iterable ``elements``.

        `I` is an order ideal if, for any `x` in `I` and `y` such that
        `y \le x`, then `y` is in `I`. This is also called lower set or
        downset.

        EXAMPLES::

            sage: P = Poset((divisors(1000), attrcall("divides")))
            sage: P.order_ideal_plot([20, 25])
            Graphics object consisting of 41 graphics primitives

        TESTS::

            sage: P = Poset()  # Test empty poset
            sage: P.order_ideal_plot([])
            Graphics object consisting of 0 graphics primitives
            sage: C = posets.ChainPoset(5)
            sage: C.order_ideal_plot([])
            Graphics object consisting of 10 graphics primitives
        """
        order_ideal = self.order_ideal(elements)
        order_filer = self.order_filter(self.order_ideal_complement_generators(order_ideal))
        order_ideal_color_dictionary = {}
        order_ideal_color_dictionary['green'] = order_ideal
        order_ideal_color_dictionary['red'] = order_filer
        return self.plot(element_colors=order_ideal_color_dictionary)

    def interval(self, x, y):
        r"""
        Return a list of the elements `z` such that `x \le z \le y`.

        INPUT:

        - ``x`` -- any element of the poset

        - ``y`` -- any element of the poset

        EXAMPLES::

            sage: uc = [[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]]
            sage: dag = DiGraph(dict(zip(range(len(uc)),uc)))
            sage: P = Poset(dag)
            sage: I = set(map(P,[2,5,6,4,7]))
            sage: I == set(P.interval(2,7))
            True

        ::

            sage: dg = DiGraph({"a":["b","c"], "b":["d"], "c":["d"]})
            sage: P = Poset(dg, facade = False)
            sage: P.interval("a","d")
            [a, b, c, d]
        """
        return [self._vertex_to_element(w)
                for w in self._hasse_diagram.interval(
                        self._element_to_vertex(x), self._element_to_vertex(y))]

    def closed_interval(self, x, y):
        r"""
        Return the list of elements `z` such that `x \le z \le y` in the poset.

        EXAMPLES::

            sage: P = Poset((divisors(1000), attrcall("divides")))
            sage: P.closed_interval(2, 100)
            [2, 4, 10, 20, 50, 100]

        .. SEEALSO::

            :meth:`open_interval`

        TESTS::

            sage: C = posets.ChainPoset(10)
            sage: C.closed_interval(3, 3)
            [3]
            sage: C.closed_interval(8, 5)
            []
            sage: A = posets.AntichainPoset(10)
            sage: A.closed_interval(3, 7)
            []
        """
        return [self._vertex_to_element(_) for _ in self._hasse_diagram.interval(
                self._element_to_vertex(x), self._element_to_vertex(y))]

    def open_interval(self, x, y):
        """
        Return the list of elements `z` such that `x < z < y` in the poset.

        EXAMPLES::

            sage: P = Poset((divisors(1000), attrcall("divides")))
            sage: P.open_interval(2, 100)
            [4, 10, 20, 50]

        .. SEEALSO::

            :meth:`closed_interval`

        TESTS::

            sage: C = posets.ChainPoset(10)
            sage: C.open_interval(3, 3)
            []
            sage: C.open_interval(3, 4)
            []
            sage: C.open_interval(7, 3)
            []
            sage: A = posets.AntichainPoset(10)
            sage: A.open_interval(3, 7)
            []
        """
        return [self._vertex_to_element(_) for _ in self._hasse_diagram.open_interval(
                self._element_to_vertex(x), self._element_to_vertex(y))]

    def comparability_graph(self):
        r"""
        Return the comparability graph of the poset.

        The comparability graph is an undirected graph where vertices
        are the elements of the poset and there is an edge between two
        vertices if they are comparable in the poset.

        See :wikipedia:`Comparability_graph`

        EXAMPLES::

            sage: Y = Poset({1: [2], 2: [3, 4]})
            sage: g = Y.comparability_graph(); g
            Comparability graph on 4 vertices
            sage: Y.compare_elements(1, 3) is not None
            True
            sage: g.has_edge(1, 3)
            True

        .. SEEALSO:: :meth:`incomparability_graph`, :mod:`sage.graphs.comparability`

        TESTS::

            sage: Poset().comparability_graph()
            Comparability graph on 0 vertices

            sage: C4 = posets.ChainPoset(4)
            sage: C4.comparability_graph().is_isomorphic(graphs.CompleteGraph(4))
            True

            sage: A4 = posets.AntichainPoset(4)
            sage: A4.comparability_graph().is_isomorphic(Graph(4))
            True
        """
        G = self.hasse_diagram().transitive_closure().to_undirected()
        G.rename('Comparability graph on %s vertices' % self.cardinality())
        return G

    def incomparability_graph(self):
        r"""
        Return the incomparability graph of the poset.

        This is the complement of the comparability graph, i.e. an
        undirected graph where vertices are the elements of the poset
        and there is an edge between vertices if they are not
        comparable in the poset.

        EXAMPLES::

            sage: Y = Poset({1: [2], 2: [3, 4]})
            sage: g = Y.incomparability_graph(); g
            Incomparability graph on 4 vertices
            sage: Y.compare_elements(1, 3) is not None
            True
            sage: g.has_edge(1, 3)
            False

        .. SEEALSO:: :meth:`comparability_graph`

        TESTS::

            sage: Poset().incomparability_graph()
            Incomparability graph on 0 vertices

            sage: C4 = posets.ChainPoset(4)
            sage: C4.incomparability_graph().is_isomorphic(Graph(4))
            True

            sage: A4 = posets.AntichainPoset(4)
            sage: A4.incomparability_graph().is_isomorphic(graphs.CompleteGraph(4))
            True
        """
        G = self.comparability_graph().complement()
        G.rename('Incomparability graph on %s vertices' % self.cardinality())
        return G

    def linear_extensions_graph(self):
        r"""
        Return the linear extensions graph of the poset.

        Vertices of the graph are linear extensions of the poset.
        Two vertices are connected by an edge if the linear extensions
        differ by only one adjacent transposition.

        EXAMPLES::

            sage: N = Poset({1: [3, 4], 2: [4]})
            sage: G = N.linear_extensions_graph(); G
            Graph on 5 vertices
            sage: G.neighbors(N.linear_extension([1,2,3,4]))
            [[2, 1, 3, 4], [1, 3, 2, 4], [1, 2, 4, 3]]

            sage: chevron = Poset({1: [2, 6], 2: [3], 4: [3, 5], 6: [5]})
            sage: G = chevron.linear_extensions_graph(); G
            Graph on 22 vertices
            sage: G.size()
            36

        TESTS::

            sage: Poset().linear_extensions_graph()
            Graph on 1 vertex

            sage: A4 = posets.AntichainPoset(4)
            sage: G = A4.linear_extensions_graph()
            sage: G.is_regular()
            True
        """
        from sage.graphs.graph import Graph
        # Direct implementation, no optimizations
        L = list(self.linear_extensions())
        G = Graph()
        G.add_vertices(L)
        for i in range(len(L)):
            for j in range(i):
                tmp = [x != y for x, y in zip(L[i], L[j])]
                if tmp.count(True) == 2 and tmp[tmp.index(True) + 1]:
                    G.add_edge(L[i], L[j])
        return G

    def maximal_antichains(self):
        r"""
        Return the maximal antichains of the poset.

        An antichain `a` of poset `P` is *maximal* if there is
        no element `e \in P \setminus a` such that `a \cup \{e\}`
        is an antichain.

        EXAMPLES::

            sage: P = Poset({'a':['b', 'c'], 'b':['d','e']})
            sage: [sorted(anti) for anti in P.maximal_antichains()]
            [['a'], ['b', 'c'], ['c', 'd', 'e']]

            sage: posets.PentagonPoset().maximal_antichains()
            [[0], [1, 2], [1, 3], [4]]

        .. SEEALSO:: :meth:`antichains`, :meth:`maximal_chains`
        """
        # Maximal antichains are maximum cliques on incomparability graph.
        return self.incomparability_graph().cliques_maximal()

    def maximal_chains(self, partial=None):
        """
        Return all maximal chains of this poset.

        Each chain is listed in increasing order.

        INPUT:

        - ``partial`` -- list (optional); if present, find all maximal
          chains starting with the elements in partial

        This is used in constructing the order complex for the poset.

        EXAMPLES::

            sage: P = posets.BooleanLattice(3)
            sage: P.maximal_chains()
            [[0, 1, 3, 7], [0, 1, 5, 7], [0, 2, 3, 7], [0, 2, 6, 7], [0, 4, 5, 7], [0, 4, 6, 7]]
            sage: P.maximal_chains(partial=[0,2])
            [[0, 2, 3, 7], [0, 2, 6, 7]]
            sage: Q = posets.ChainPoset(6)
            sage: Q.maximal_chains()
            [[0, 1, 2, 3, 4, 5]]

        .. SEEALSO:: :meth:`maximal_antichains`, :meth:`chains`
        """
        return list(self.maximal_chains_iterator(partial=partial))

    def maximal_chains_iterator(self, partial=None):
        """
        Return an iterator over maximal chains.

        Each chain is listed in increasing order.

        INPUT:

        - ``partial`` -- list (optional); if present, yield all maximal
          chains starting with the elements in partial

        EXAMPLES::

            sage: P = posets.BooleanLattice(3)
            sage: it = P.maximal_chains_iterator()
            sage: next(it)
            [0, 1, 3, 7]

        .. SEEALSO::

            :meth:`antichains_iterator`
        """
        if not partial:
            start = self.minimal_elements()
            partial = []
        else:
            start = self.upper_covers(partial[-1])
        if not start:
            yield partial
        elif len(start) == 1:
            yield from self.maximal_chains_iterator(partial=partial + start)
        else:
            parts = (partial + [x] for x in start)
            for new in parts:
                yield from self.maximal_chains_iterator(partial=new)

    def maximal_chain_length(self):
        """
        Return the maximum length of a maximal chain in the poset.

        The length here is the number of vertices.

        EXAMPLES::

            sage: P = posets.TamariLattice(5)
            sage: P.maximal_chain_length()
            11

        TESTS::

            sage: Poset().maximal_chain_length()
            0

        .. SEEALSO:: :meth:`maximal_chains`, :meth:`maximal_chains_iterator`
        """
        if not self.cardinality():
            return 0
        store = {}
        for x in self:
            below = self.lower_covers(x)
            if not below:
                store[x] = 1
            else:
                store[x] = 1 + max(store[y] for y in below)
        return max(store.values())

    def order_complex(self, on_ints=False):
        r"""
        Return the order complex associated to this poset.

        The order complex is the simplicial complex with vertices equal
        to the elements of the poset, and faces given by the chains.

        INPUT:

        - ``on_ints`` -- a boolean (default: ``False``)

        OUTPUT:

        an order complex of type :class:`SimplicialComplex`

        EXAMPLES::

            sage: P = posets.BooleanLattice(3)
            sage: S = P.order_complex(); S
            Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and 6 facets
            sage: S.f_vector()
            [1, 8, 19, 18, 6]
            sage: S.homology()      # S is contractible
            {0: 0, 1: 0, 2: 0, 3: 0}
            sage: Q = P.subposet([1,2,3,4,5,6])
            sage: Q.order_complex().homology()    # a circle
            {0: 0, 1: Z}

            sage: P = Poset((divisors(15), attrcall("divides")), facade = True)
            sage: P.order_complex()
            Simplicial complex with vertex set (1, 3, 5, 15) and facets {(1, 3, 15), (1, 5, 15)}

        If ``on_ints``, then the elements of the poset are labelled
        `0,1,\dots` in the chain complex::

            sage: P.order_complex(on_ints=True)
            Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1, 3), (0, 2, 3)}
        """
        from sage.topology.simplicial_complex import SimplicialComplex
        L = self.list()
        if on_ints:
            iso = dict([(L[i], i) for i in range(len(L))])

        facets = []
        for f in self.maximal_chains_iterator():
            # TODO: factor out the logic for on_ints / facade / ...
            # We will want to do similar things elsewhere
            if on_ints:
                facets.append([iso[a] for a in f])
            elif self._is_facade:
                facets.append([a for a in f])
            else:
                facets.append([a.element for a in f])

        return SimplicialComplex(facets, maximality_check=False)

    def order_polytope(self):
        r"""
        Return the order polytope of the poset ``self``.

        The order polytope of a finite poset `P` is defined as the subset
        of `\RR^P` consisting of all maps `x : P \to \RR` satisfying

        .. MATH::

            0 \leq x(p) \leq 1 \mbox{ for all } p \in P,

        and

        .. MATH::

            x(p) \leq x(q) \mbox{ for all } p, q \in P
            \mbox{ satisfying } p < q.

        This polytope was defined and studied in [St1986]_.

        EXAMPLES::

            sage: P = posets.AntichainPoset(3)
            sage: Q = P.order_polytope();Q
            A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices
            sage: P = posets.PentagonPoset()
            sage: Q = P.order_polytope();Q
            A 5-dimensional polyhedron in ZZ^5 defined as the convex hull of 8 vertices

            sage: P = Poset([[1,2,3],[[1,2],[1,3]]])
            sage: Q = P.order_polytope()
            sage: Q.contains((1,0,0))
            False
            sage: Q.contains((0,1,1))
            True
        """
        from sage.geometry.polyhedron.constructor import Polyhedron
        ineqs = [[0] + [ZZ(j == v) - ZZ(j == u) for j in self]
                 for u, v, w in self.hasse_diagram().edges()]
        for i in self.maximal_elements():
            ineqs += [[1] + [-ZZ(j == i) for j in self]]
        for i in self.minimal_elements():
            ineqs += [[0] + [ZZ(j == i) for j in self]]
        return Polyhedron(ieqs=ineqs, base_ring=ZZ)

    def chain_polytope(self):
        r"""
        Return the chain polytope of the poset ``self``.

        The chain polytope of a finite poset `P` is defined as the subset
        of `\RR^P` consisting of all maps `x : P \to \RR` satisfying

        .. MATH::

            x(p) \geq 0 \mbox{ for all } p \in P,

        and

        .. MATH::

            x(p_1) + x(p_2) + \ldots + x(p_k) \leq 1
            \mbox{ for all chains } p_1 < p_2 < \ldots < p_k
            \mbox{ in } P.

        This polytope was defined and studied in [St1986]_.

        EXAMPLES::

            sage: P = posets.AntichainPoset(3)
            sage: Q = P.chain_polytope();Q
            A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices
            sage: P = posets.PentagonPoset()
            sage: Q = P.chain_polytope();Q
            A 5-dimensional polyhedron in ZZ^5 defined as the convex hull of 8 vertices
        """
        from sage.geometry.polyhedron.constructor import Polyhedron
        ineqs = [[1] + [-ZZ(j in chain) for j in self]
               for chain in self.maximal_chains_iterator()]
        for i in self:
            ineqs += [[0] + [ZZ(j == i) for j in self]]
        return Polyhedron(ieqs=ineqs, base_ring=ZZ)

    def zeta_polynomial(self):
        r"""
        Return the zeta polynomial of the poset.

        The zeta polynomial of a poset is the unique polynomial `Z(q)`
        such that for every integer `m > 1`, `Z(m)` is the number of
        weakly increasing sequences `x_1 \leq x_2 \leq \dots \leq x_{m-1}`
        of elements of the poset.

        The polynomial `Z(q)` is integral-valued, but generally does not
        have integer coefficients. It can be computed as

        .. MATH::

            Z(q) = \sum_{k \geq 1} \dbinom{q-2}{k-1} c_k,

        where `c_k` is the number of all chains of length `k` in the
        poset.

        For more information, see section 3.12 of [EnumComb1]_.

        In particular, `Z(2)` is the number of vertices and `Z(3)` is
        the number of intervals.

        EXAMPLES::

            sage: posets.ChainPoset(2).zeta_polynomial()
            q
            sage: posets.ChainPoset(3).zeta_polynomial()
            1/2*q^2 + 1/2*q

            sage: P = posets.PentagonPoset()
            sage: P.zeta_polynomial()
            1/6*q^3 + q^2 - 1/6*q

            sage: P = posets.DiamondPoset(5)
            sage: P.zeta_polynomial()
            3/2*q^2 - 1/2*q

        TESTS:

        Checking the simplest cases::

            sage: Poset({}).zeta_polynomial()
            0
            sage: Poset({1: []}).zeta_polynomial()
            1
            sage: Poset({1: [], 2: []}).zeta_polynomial()
            2
            sage: parent(_)
            Univariate Polynomial Ring in q over Rational Field
        """
        R = PolynomialRing(QQ, 'q')
        q = R.gen()

        g = self.chain_polynomial()
        n = g.degree()
        f = R(g[max(n, 1)])
        while n > 1:
            f = (q - n) * f
            n -= 1
            f = g[n] + f / n
        return f

    def f_polynomial(self):
        r"""
        Return the `f`-polynomial of the poset.

        The poset is expected to be bounded.

        This is the `f`-polynomial of the order complex of the poset
        minus its bounds.

        The coefficient of `q^i` is the number of chains of
        `i+1` elements containing both bounds of the poset.

        .. note::

            This is slightly different from the ``fPolynomial``
            method in Macaulay2.

        EXAMPLES::

            sage: P = posets.DiamondPoset(5)
            sage: P.f_polynomial()
            3*q^2 + q

            sage: P = Poset({1: [2, 3], 2: [4], 3: [5], 4: [6], 5: [7], 6: [7]})
            sage: P.f_polynomial()
            q^4 + 4*q^3 + 5*q^2 + q

        .. SEEALSO::

            :meth:`is_bounded`, :meth:`h_polynomial`, :meth:`order_complex`,
            :meth:`sage.topology.cell_complex.GenericCellComplex.f_vector`

        TESTS::

            sage: P = Poset({2: []})
            sage: P.f_polynomial()
            1
        """
        q = polygen(ZZ, 'q')
        one = q.parent().one()
        hasse = self._hasse_diagram
        if len(hasse) == 1:
            return one
        maxi = hasse.top()
        mini = hasse.bottom()
        if mini is None or maxi is None:
            raise ValueError("the poset is not bounded")

        hasse_size = hasse.cardinality()
        chain_polys = [0] * hasse_size
        # chain_polys[i] will be the generating function for the
        # chains with topmost vertex i (in the labelling of the
        # Hasse diagram).
        for i in range(1, hasse_size - 1):
            chain_polys[i] = q + sum(q * chain_polys[j]
                                     for j in hasse.principal_order_ideal(i)
                                     if j)
        return q + q * sum(chain_polys)

    def h_polynomial(self):
        r"""
        Return the `h`-polynomial of a bounded poset ``self``.

        This is the `h`-polynomial of the order complex of the poset
        minus its bounds.

        This is related to the `f`-polynomial by a simple change
        of variables:

        .. MATH::

            h(q) = (1-q)^{\deg f} f \left( \frac{q}{1-q} \right),

        where `f` and `h` denote the `f`-polynomial and the
        `h`-polynomial, respectively.

        See :wikipedia:`h-vector`.

        .. WARNING::

            This is slightly different from the ``hPolynomial``
            method in Macaulay2.

        EXAMPLES::

            sage: P = posets.AntichainPoset(3).order_ideals_lattice()
            sage: P.h_polynomial()
            q^3 + 4*q^2 + q
            sage: P = posets.DiamondPoset(5)
            sage: P.h_polynomial()
            2*q^2 + q
            sage: P = Poset({1: []})
            sage: P.h_polynomial()
            1

        .. SEEALSO::

            :meth:`is_bounded`, :meth:`f_polynomial`, :meth:`order_complex`,
            :meth:`sage.topology.simplicial_complex.SimplicialComplex.h_vector`
        """
        q = polygen(ZZ, 'q')
        hasse = self._hasse_diagram
        if len(hasse) == 1:
            return q.parent().one()
        maxi = hasse.top()
        mini = hasse.bottom()
        if (mini is None) or (maxi is None):
            raise ValueError("the poset is not bounded")
        f = sum(q**(len(ch)) for ch in hasse.chains(exclude=[mini, maxi]))
        d = f.degree()
        f = (1 - q)**d * q * f(q=q / (1 - q))
        return q.parent(f)

    def flag_f_polynomial(self):
        r"""
        Return the flag `f`-polynomial of the poset.

        The poset is expected to be bounded and ranked.

        This is the sum, over all chains containing both bounds,
        of a monomial encoding the ranks of the elements of the chain.

        More precisely, if `P` is a bounded ranked poset, then the
        flag `f`-polynomial of `P` is defined as the polynomial

        .. MATH::

            \sum_{\substack{p_0 < p_1 < \ldots < p_k, \\
                            p_0 = \min P, \ p_k = \max P}}
            x_{\rho(p_1)} x_{\rho(p_2)} \cdots x_{\rho(p_k)}
            \in \ZZ[x_1, x_2, \cdots, x_n],

        where `\min P` and `\max P` are (respectively) the minimum and
        the maximum of `P`, where `\rho` is the rank function of `P`
        (normalized to satisfy `\rho(\min P) = 0`), and where
        `n` is the rank of `\max P`. (Note that the indeterminate
        `x_0` does not actually appear in the polynomial.)

        For technical reasons, the polynomial is returned in the
        slightly larger ring `\ZZ[x_0, x_1, x_2, \cdots, x_{n+1}]` by
        this method.

        See :wikipedia:`h-vector`.

        EXAMPLES::

            sage: P = posets.DiamondPoset(5)
            sage: P.flag_f_polynomial()
            3*x1*x2 + x2

            sage: P = Poset({1: [2, 3], 2: [4], 3: [5], 4: [6], 5: [6]})
            sage: fl = P.flag_f_polynomial(); fl
            2*x1*x2*x3 + 2*x1*x3 + 2*x2*x3 + x3
            sage: q = polygen(ZZ,'q')
            sage: fl(q,q,q,q) == P.f_polynomial()
            True

            sage: P = Poset({1: [2, 3, 4], 2: [5], 3: [5], 4: [5], 5: [6]})
            sage: P.flag_f_polynomial()
            3*x1*x2*x3 + 3*x1*x3 + x2*x3 + x3

        .. SEEALSO:: :meth:`is_bounded`, :meth:`flag_h_polynomial`

        TESTS::

            sage: P = Poset({2: [3]})
            sage: P.flag_f_polynomial()
            x1

            sage: P = Poset({2: []})
            sage: P.flag_f_polynomial()
            1
        """
        hasse = self._hasse_diagram
        maxi = hasse.top()
        mini = hasse.bottom()
        if (mini is None) or (maxi is None):
            raise ValueError("the poset is not bounded")
        rk = hasse.rank_function()
        if rk is None:
            raise ValueError("the poset is not ranked")
        n = rk(maxi)
        if n == 0:
            return PolynomialRing(ZZ, 'x', 1).one()
        anneau = PolynomialRing(ZZ, 'x', n+1)
        x = anneau.gens()
        return x[n] * sum(prod(x[rk(i)] for i in ch) for ch in hasse.chains(exclude=[mini, maxi]))

    def flag_h_polynomial(self):
        r"""
        Return the flag `h`-polynomial of the poset.

        The poset is expected to be bounded and ranked.

        If `P` is a bounded ranked poset whose maximal element has
        rank `n` (where the minimal element is set to have rank `0`),
        then the flag `h`-polynomial of `P` is defined as the
        polynomial

        .. MATH::

            \prod_{k=1}^n (1-x_k) \cdot f \left(\frac{x_1}{1-x_1},
            \frac{x_2}{1-x_2}, \cdots, \frac{x_n}{1-x_n}\right)
            \in \ZZ[x_1, x_2, \cdots, x_n],

        where `f` is the flag `f`-polynomial of `P` (see
        :meth:`flag_f_polynomial`).

        For technical reasons, the polynomial is returned in the
        slightly larger ring `\QQ[x_0, x_1, x_2, \cdots, x_{n+1}]` by
        this method.

        See :wikipedia:`h-vector`.

        EXAMPLES::

            sage: P = posets.DiamondPoset(5)
            sage: P.flag_h_polynomial()
            2*x1*x2 + x2

            sage: P = Poset({1: [2, 3], 2: [4], 3: [5], 4: [6], 5: [6]})
            sage: fl = P.flag_h_polynomial(); fl
            -x1*x2*x3 + x1*x3 + x2*x3 + x3
            sage: q = polygen(ZZ,'q')
            sage: fl(q,q,q,q) == P.h_polynomial()
            True

            sage: P = Poset({1: [2, 3, 4], 2: [5], 3: [5], 4: [5], 5: [6]})
            sage: P.flag_h_polynomial()
            2*x1*x3 + x3

            sage: P = posets.ChainPoset(4)
            sage: P.flag_h_polynomial()
            x3

        .. SEEALSO:: :meth:`is_bounded`, :meth:`flag_f_polynomial`

        TESTS::

            sage: P = Poset({2: [3]})
            sage: P.flag_h_polynomial()
            x1

            sage: P = Poset({2: []})
            sage: P.flag_h_polynomial()
            1
        """
        hasse = self._hasse_diagram
        maxi = hasse.top()
        mini = hasse.bottom()
        if (mini is None) or (maxi is None):
            raise ValueError("the poset is not bounded")
        rk = hasse.rank_function()
        if rk is None:
            raise ValueError("the poset is not ranked")
        n = rk(maxi)
        if n == 0:
            return PolynomialRing(QQ, 'x', 1).one()
        anneau = PolynomialRing(QQ, 'x', n + 1)
        x = anneau.gens()
        return prod(1 - x[k] for k in range(1, n)) * x[n] \
            * sum(prod(x[rk(i)] / (1 - x[rk(i)]) for i in ch)
                  for ch in hasse.chains(exclude=[mini, maxi]))

    def characteristic_polynomial(self):
        r"""
        Return the characteristic polynomial of the poset.

        The poset is expected to be graded and have a bottom
        element.

        If `P` is a graded poset with rank `n` and a unique minimal
        element `\hat{0}`, then the characteristic polynomial of
        `P` is defined to be

        .. MATH::

            \sum_{x \in P} \mu(\hat{0}, x) q^{n-\rho(x)} \in \ZZ[q],

        where `\rho` is the rank function, and `\mu` is the Möbius
        function of `P`.

        See section 3.10 of [EnumComb1]_.

        EXAMPLES::

            sage: P = posets.DiamondPoset(5)
            sage: P.characteristic_polynomial()
            q^2 - 3*q + 2

            sage: P = Poset({1: [2, 3], 2: [4], 3: [5], 4: [6], 5: [6], 6: [7]})
            sage: P.characteristic_polynomial()
            q^4 - 2*q^3 + q

        TESTS::

            sage: P = Poset({1: []})
            sage: P.characteristic_polynomial()
            1
        """
        hasse = self._hasse_diagram
        rk = hasse.rank_function()
        if not self.is_graded():
            raise ValueError("the poset is not graded")
        if not self.has_bottom():
            raise ValueError("the poset has not a bottom element")
        n = rk(hasse.maximal_elements()[0])
        x0 = hasse.minimal_elements()[0]
        q = polygen(ZZ, 'q')
        return sum(hasse.moebius_function(x0, x) * q**(n - rk(x)) for x in hasse)

    def chain_polynomial(self):
        """
        Return the chain polynomial of the poset.

        The coefficient of `q^k` is the number of chains of `k`
        elements in the poset. List of coefficients of this polynomial
        is also called a *f-vector* of the poset.

        .. note::

            This is not what has been called the chain polynomial
            in [St1986]_. The latter is identical with the order
            polynomial in SageMath (:meth:`order_polynomial`).

        .. SEEALSO:: :meth:`f_polynomial`, :meth:`order_polynomial`

        EXAMPLES::

            sage: P = posets.ChainPoset(3)
            sage: t = P.chain_polynomial(); t
            q^3 + 3*q^2 + 3*q + 1
            sage: t(1) == len(list(P.chains()))
            True

            sage: P = posets.BooleanLattice(3)
            sage: P.chain_polynomial()
            6*q^4 + 18*q^3 + 19*q^2 + 8*q + 1

            sage: P = posets.AntichainPoset(5)
            sage: P.chain_polynomial()
            5*q + 1

        TESTS::

            sage: P = Poset()
            sage: P.chain_polynomial()
            1
            sage: parent(P.chain_polynomial())
            Univariate Polynomial Ring in q over Integer Ring

            sage: R = Poset({1: []})
            sage: R.chain_polynomial()
            q + 1
        """
        hasse = self._hasse_diagram
        q = polygen(ZZ, 'q')
        one = q.parent().one()
        hasse_size = hasse.cardinality()
        chain_polys = [0] * hasse_size
        # chain_polys[i] will be the generating function for the
        # chains with topmost vertex i (in the labelling of the
        # Hasse diagram).
        for i in range(hasse_size):
            chain_polys[i] = q + sum(q * chain_polys[j]
                                     for j in hasse.principal_order_ideal(i))
        return one + sum(chain_polys)

    def order_polynomial(self):
        r"""
        Return the order polynomial of the poset.

        The order polynomial `\Omega_P(q)` of a poset `P` is defined
        as the unique polynomial `S` such that for each integer
        `m \geq 1`, `S(m)` is the number of order-preserving maps
        from `P` to `\{1,\ldots,m\}`.

        See sections 3.12 and 3.15 of [EnumComb1]_, and also
        [St1986]_.

        EXAMPLES::

            sage: P = posets.AntichainPoset(3)
            sage: P.order_polynomial()
            q^3

            sage: P = posets.ChainPoset(3)
            sage: f = P.order_polynomial(); f
            1/6*q^3 + 1/2*q^2 + 1/3*q
            sage: [f(i) for i in range(4)]
            [0, 1, 4, 10]

        .. SEEALSO:: :meth:`order_polytope`
        """
        return self.order_ideals_lattice(as_ideals=False).zeta_polynomial()

    def degree_polynomial(self):
        r"""
        Return the generating polynomial of degrees of vertices in ``self``.

        This is the sum

        .. MATH::

            \sum_{v \in P} x^{\operatorname{in}(v)} y^{\operatorname{out}(v)},

        where ``in(v)`` and ``out(v)`` are the number of incoming and
        outgoing edges at vertex `v` in the Hasse diagram of `P`.

        Because this polynomial is multiplicative for Cartesian
        product of posets, it is useful to help see if the poset can
        be isomorphic to a Cartesian product.

        EXAMPLES::

            sage: P = posets.PentagonPoset()
            sage: P.degree_polynomial()
            x^2 + 3*x*y + y^2

            sage: P = posets.BooleanLattice(4)
            sage: P.degree_polynomial().factor()
            (x + y)^4

        .. SEEALSO::

            :meth:`cardinality` for the value at `(x, y) = (1, 1)`
        """
        return self._hasse_diagram.degree_polynomial()

    def promotion(self, i=1):
        r"""
        Compute the (extended) promotion on the linear extension
        of the poset ``self``.

        INPUT:

        - ``i`` -- an integer between `1` and `n` (default: `1`)

        OUTPUT:

        - an isomorphic poset, with the same default linear extension

        The extended promotion is defined on a poset ``self`` of size
        `n` by applying the promotion operator `\tau_i \tau_{i+1}
        \cdots \tau_{n-1}` to the default linear extension `\pi` of ``self``
        (see :meth:`~sage.combinat.posets.linear_extensions.LinearExtensionOfPoset.promotion`),
        and relabeling ``self`` accordingly. For more details see [Stan2009]_.

        When the elements of the poset ``self`` are labelled by
        `\{1,2,\ldots,n\}`, the linear extension is the identity, and
        `i=1`, the above algorithm corresponds to the promotion
        operator on posets defined by Schützenberger as
        follows. Remove `1` from ``self`` and replace it by the
        minimum `j` of all labels covering `1` in the poset. Then,
        remove `j` and replace it by the minimum of all labels
        covering `j`, and so on.  This process ends when a label is a
        local maximum. Place the label `n+1` at this vertex.  Finally,
        decrease all labels by `1`.

        EXAMPLES::

            sage: P = Poset(([1,2], [[1,2]]), linear_extension=True, facade=False)
            sage: P.promotion()
            Finite poset containing 2 elements with distinguished linear extension
            sage: P == P.promotion()
            True

            sage: P = Poset(([1,2,3,4,5,6,7], [[1,2],[1,4],[2,3],[2,5],[3,6],[4,7],[5,6]]))
            sage: P.list()
            [1, 2, 3, 5, 6, 4, 7]
            sage: Q = P.promotion(4); Q
            Finite poset containing 7 elements with distinguished linear extension
            sage: Q.cover_relations()
            [[1, 2], [1, 6], [2, 3], [2, 5], [3, 7], [5, 7], [6, 4]]

        Note that if one wants to obtain the promotion defined by
        Schützenberger's algorithm directly on the poset, one needs
        to make sure the linear extension is the identity::

            sage: P = P.with_linear_extension([1,2,3,4,5,6,7])
            sage: P.list()
            [1, 2, 3, 4, 5, 6, 7]
            sage: Q = P.promotion(4); Q
            Finite poset containing 7 elements with distinguished linear extension
            sage: Q.cover_relations()
            [[1, 2], [1, 6], [2, 3], [2, 4], [3, 5], [4, 5], [6, 7]]
            sage: Q = P.promotion()
            sage: Q.cover_relations()
            [[1, 2], [1, 3], [2, 4], [2, 5], [3, 6], [4, 7], [5, 7]]

        Here is an example for a poset not labelled by `\{1, 2, \ldots, n\}`::

            sage: P = Poset((divisors(30), attrcall("divides")), linear_extension=True)
            sage: P.list()
            [1, 2, 3, 5, 6, 10, 15, 30]
            sage: P.cover_relations()
            [[1, 2], [1, 3], [1, 5], [2, 6], [2, 10], [3, 6], [3, 15],
             [5, 10], [5, 15], [6, 30], [10, 30], [15, 30]]
            sage: Q = P.promotion(4); Q
            Finite poset containing 8 elements with distinguished linear extension
            sage: Q.cover_relations()
            [[1, 2], [1, 3], [1, 6], [2, 5], [2, 15], [3, 5], [3, 10],
             [5, 30], [6, 10], [6, 15], [10, 30], [15, 30]]

        .. SEEALSO::

            - :meth:`linear_extension`
            - :meth:`with_linear_extension` and the ``linear_extension`` option of :func:`Poset`
            - :meth:`~sage.combinat.posets.linear_extensions.LinearExtensionOfPoset.promotion`
            - :meth:`evacuation`

        AUTHOR:

        - Anne Schilling (2012-02-18)
        """
        return self.linear_extension().promotion(i).to_poset()

    def evacuation(self):
        r"""
        Compute evacuation on the linear extension associated
        to the poset ``self``.

        OUTPUT:

        - an isomorphic poset, with the same default linear extension

        Evacuation is defined on a poset ``self`` of size `n` by
        applying the evacuation operator
        `(\tau_1 \cdots \tau_{n-1}) (\tau_1 \cdots \tau_{n-2}) \cdots (\tau_1)`,
        to the default linear extension `\pi` of ``self``
        (see :meth:`~sage.combinat.posets.linear_extensions.LinearExtensionOfPoset.evacuation`),
        and relabeling ``self`` accordingly. For more details see [Stan2009]_.

        EXAMPLES::

            sage: P = Poset(([1,2], [[1,2]]), linear_extension=True, facade=False)
            sage: P.evacuation()
            Finite poset containing 2 elements with distinguished linear extension
            sage: P.evacuation() == P
            True

            sage: P = Poset(([1,2,3,4,5,6,7], [[1,2],[1,4],[2,3],[2,5],[3,6],[4,7],[5,6]]), linear_extension=True, facade=False)
            sage: P.list()
            [1, 2, 3, 4, 5, 6, 7]
            sage: Q = P.evacuation(); Q
            Finite poset containing 7 elements with distinguished linear extension
            sage: Q.cover_relations()
            [[1, 2], [1, 3], [2, 5], [3, 4], [3, 6], [4, 7], [6, 7]]

        Note that the results depend on the linear extension associated
        to the poset::

            sage: P = Poset(([1,2,3,4,5,6,7], [[1,2],[1,4],[2,3],[2,5],[3,6],[4,7],[5,6]]))
            sage: P.list()
            [1, 2, 3, 5, 6, 4, 7]
            sage: Q = P.evacuation(); Q
            Finite poset containing 7 elements with distinguished linear extension
            sage: Q.cover_relations()
            [[1, 2], [1, 5], [2, 3], [5, 6], [5, 4], [6, 7], [4, 7]]

        Here is an example of a poset where the elements are not labelled
        by `\{1,2,\ldots,n\}`::

            sage: P = Poset((divisors(15), attrcall("divides")), linear_extension = True)
            sage: P.list()
            [1, 3, 5, 15]
            sage: Q = P.evacuation(); Q
            Finite poset containing 4 elements with distinguished linear extension
            sage: Q.cover_relations()
            [[1, 3], [1, 5], [3, 15], [5, 15]]

        .. SEEALSO::

            - :meth:`linear_extension`
            - :meth:`with_linear_extension` and the ``linear_extension`` option of :func:`Poset`
            - :meth:`~sage.combinat.posets.linear_extensions.LinearExtensionOfPoset.evacuation`
            - :meth:`promotion`

        AUTHOR:

        - Anne Schilling (2012-02-18)
        """
        return self.linear_extension().evacuation().to_poset()

    def is_rank_symmetric(self) -> bool:
        r"""
        Return ``True`` if the poset is rank symmetric, and ``False``
        otherwise.

        The poset is expected to be graded and connected.

        A poset of rank `h` (maximal chains have `h+1` elements) is rank
        symmetric if the number of elements are equal in ranks `i` and
        `h-i` for every `i` in `0, 1, \ldots, h`.

        EXAMPLES::

            sage: P = Poset({1:[3, 4, 5], 2:[3, 4, 5], 3:[6], 4:[7], 5:[7]})
            sage: P.is_rank_symmetric()
            True
            sage: P = Poset({1:[2], 2:[3, 4], 3:[5], 4:[5]})
            sage: P.is_rank_symmetric()
            False

        TESTS::

            sage: Poset().is_rank_symmetric()  # Test empty poset
            True
        """
        if not self.is_connected():
            raise ValueError("the poset is not connected")
        if not self.is_graded():
            raise ValueError("the poset is not graded")
        levels = self._hasse_diagram.level_sets()
        h = len(levels)
        for i in range(h // 2):
            if len(levels[i]) != len(levels[h - 1 - i]):
                return False
        return True

    def is_slender(self, certificate=False):
        r"""
        Return ``True`` if the poset is slender, and ``False`` otherwise.

        A finite graded poset is *slender* if every rank 2
        interval contains three or four elements, as defined in
        [Stan2009]_. (This notion of "slender" is unrelated to
        the eponymous notion defined by Graetzer and Kelly in
        "The Free $\mathfrak{m}$-Lattice on the Poset $H$",
        Order 1 (1984), 47--65.)

        This function *does not* check if the poset is graded or not.
        Instead it just returns ``True`` if the poset does not contain
        5 distinct elements `x`, `y`, `a`, `b` and `c` such that
        `x \lessdot a,b,c \lessdot y` where `\lessdot` is the covering
        relation.

        INPUT:

        - ``certificate`` -- (default: ``False``) whether to return
          a certificate

        OUTPUT:

        - If ``certificate=True`` return either ``(True, None)`` or
          ``(False, (a, b))`` so that the interval `[a, b]` has at
          least five elements. If ``certificate=False`` return
          ``True`` or ``False``.

        EXAMPLES::

            sage: P = Poset(([1, 2, 3, 4], [[1, 2], [1, 3], [2, 4], [3, 4]]))
            sage: P.is_slender()
            True
            sage: P = Poset(([1,2,3,4,5],[[1,2],[1,3],[1,4],[2,5],[3,5],[4,5]]))
            sage: P.is_slender()
            False

            sage: W = WeylGroup(['A', 2])
            sage: G = W.bruhat_poset()
            sage: G.is_slender()
            True
            sage: W = WeylGroup(['A', 3])
            sage: G = W.bruhat_poset()
            sage: G.is_slender()
            True

            sage: P = posets.IntegerPartitions(6)
            sage: P.is_slender(certificate=True)
            (False, ((6,), (3, 2, 1)))

        TESTS::

            sage: Poset().is_slender()  # Test empty poset
            True

        Correct certificate (:trac:`22373`)::

            sage: P = Poset({0:[1,2,3],1:[4],2:[4],3:[4,5]})
            sage: P.is_slender(True)
            (False, (0, 4))
        """
        for x in self:
            d = {}
            for y in self.upper_covers(x):
                for c in self.upper_covers(y):
                    d[c] = d.get(c, 0) + 1
            for c, y in d.items():
                if y >= 3:
                    if certificate:
                        return (False, (x, c))
                    return False
        if certificate:
            return (True, None)
        return True

    def is_sperner(self):
        """
        Return ``True`` if the poset is Sperner, and ``False`` otherwise.

        The poset is expected to be ranked.

        A poset is Sperner, if no antichain is larger than the largest
        rank level (one of the sets of elements of the same rank) in
        the poset.

        See :wikipedia:`Sperner_property_of_a_partially_ordered_set`

        .. SEEALSO:: :meth:`width`, :meth:`dilworth_decomposition`

        EXAMPLES::

            sage: posets.SetPartitions(3).is_sperner()
            True

            sage: P = Poset({0:[3,4,5],1:[5],2:[5]})
            sage: P.is_sperner()
            False

        TESTS::

            sage: posets.PentagonPoset().is_sperner()
            Traceback (most recent call last):
            ...
            ValueError: the poset is not ranked

            sage: P = Poset()
            sage: P.is_sperner()
            True
        """
        if not self.is_ranked():
            raise ValueError("the poset is not ranked")
        if not self.cardinality():
            return True
        W = self.width()
        N = max(len(level) for level in self._hasse_diagram.level_sets())
        return W <= N

    def is_eulerian(self, k=None, certificate=False):
        """
        Return ``True`` if the poset is Eulerian, and ``False`` otherwise.

        The poset is expected to be graded and bounded.

        A poset is Eulerian if every non-trivial interval has the same
        number of elements of even rank as of odd rank. A poset is
        `k`-eulerian if every non-trivial interval up to rank `k`
        is Eulerian.

        See :wikipedia:`Eulerian_poset`.

        INPUT:

        - ``k``, an integer -- only check if the poset is `k`-eulerian.
          If ``None`` (the default), check if the poset is Eulerian.
        - ``certificate``, a Boolean -- (default: ``False``) whether to return
          a certificate

        OUTPUT:

        - If ``certificate=True`` return either ``True, None`` or
          ``False, (a, b)``, where the interval ``(a, b)`` is not
          Eulerian. If ``certificate=False`` return ``True`` or ``False``.

        EXAMPLES::

            sage: P = Poset({0: [1, 2, 3], 1: [4, 5], 2: [4, 6], 3: [5, 6],
            ....:            4: [7, 8], 5: [7, 8], 6: [7, 8], 7: [9], 8: [9]})
            sage: P.is_eulerian()
            True
            sage: P = Poset({0: [1, 2, 3], 1: [4, 5, 6], 2: [4, 6], 3: [5,6],
            ....:            4: [7], 5:[7], 6:[7]})
            sage: P.is_eulerian()
            False

        Canonical examples of Eulerian posets are the face lattices of
        convex polytopes::

            sage: P = polytopes.cube().face_lattice()
            sage: P.is_eulerian()
            True

        A poset that is 3- but not 4-eulerian::

            sage: P = Poset(DiGraph('MWW@_?W?@_?W??@??O@_?W?@_?W?@??O??')); P
            Finite poset containing 14 elements
            sage: P.is_eulerian(k=3)
            True
            sage: P.is_eulerian(k=4)
            False

        Getting an interval that is not Eulerian::

            sage: P = posets.DivisorLattice(12)
            sage: P.is_eulerian(certificate=True)
            (False, (1, 4))

        TESTS::

            sage: Poset().is_eulerian()
            Traceback (most recent call last):
            ...
            ValueError: the poset is not bounded

            sage: Poset({1: []}).is_eulerian()
            True

            sage: posets.PentagonPoset().is_eulerian()
            Traceback (most recent call last):
            ...
            ValueError: the poset is not graded

            sage: posets.BooleanLattice(3).is_eulerian(k=123, certificate=True)
            (True, None)
        """
        if k is not None:
            try:
                k = Integer(k)
            except TypeError:
                raise TypeError("parameter 'k' must be an integer, not {0}".format(k))
            if k <= 0:
                raise ValueError("parameter 'k' must be positive, not {0}".format(k))

        if not self.is_bounded():
            raise ValueError("the poset is not bounded")
        if not self.is_ranked():
            raise ValueError("the poset is not graded")

        n = self.cardinality()
        if n == 1:
            return True
        if k is None and not certificate and n % 2 == 1:
            return False

        H = self._hasse_diagram
        M = H.moebius_function_matrix()
        levels = H.level_sets()
        height = len(levels)
        if k is None or k > height:
            k = height

        # Every 2n -eulerian poset is always also 2n+1 -eulerian. Hence
        # we only check for even rank intervals. See for example
        # Richard Ehrenborg, k-Eulerian Posets (Order 18: 227-236, 2001)
        # http://www.ms.uky.edu/~jrge/Papers/k-Eulerian.pdf
        for rank_diff in range(2, k + 1, 2):
            for level in range(height - rank_diff):
                for i in levels[level]:
                    for j in levels[level+rank_diff]:
                        if H.is_lequal(i, j) and M[i, j] != 1:
                            if certificate:
                                return (False, (self._vertex_to_element(i),
                                                self._vertex_to_element(j)))
                            return False
        return (True, None) if certificate else True

    def is_greedy(self, certificate=False):
        """
        Return ``True`` if the poset is greedy, and ``False`` otherwise.

        A poset is *greedy* if every greedy linear extension
        has the same number of jumps.

        INPUT:

        - ``certificate`` -- (default: ``False``) whether to return
          a certificate

        OUTPUT:

        - If ``certificate=True`` return either ``(True, None)`` or
          ``(False, (A, B))`` where `A` and `B` are greedy linear extension
          so that `B` has more jumps. If ``certificate=False`` return
          ``True`` or ``False``.

        EXAMPLES:

        This is not a self-dual property::

            sage: W = Poset({1: [3, 4], 2: [4, 5]})
            sage: M = W.dual()
            sage: W.is_greedy()
            True
            sage: M.is_greedy()
            False

        Getting a certificate::

            sage: N = Poset({1: [3], 2: [3, 4]})
            sage: N.is_greedy(certificate=True)
            (False, ([1, 2, 4, 3], [2, 4, 1, 3]))

        TESTS::

            sage: Poset().is_greedy()
            True
            sage: posets.AntichainPoset(3).is_greedy()
            True
            sage: posets.ChainPoset(3).is_greedy()
            True
        """
        H = self._hasse_diagram
        N1 = H.order()-1
        it = H.greedy_linear_extensions_iterator()
        A = next(it)
        A_jumps = sum(1 for i in range(N1) if H.has_edge(A[i], A[i+1]))

        for B in it:
            B_jumps = sum(1 for i in range(N1) if H.has_edge(B[i], B[i+1]))
            if A_jumps != B_jumps:
                if certificate:
                    if A_jumps > B_jumps:
                        A, B = B, A
                    return (False,
                            (self.linear_extension([self[v] for v in A]),
                             self.linear_extension([self[v] for v in B])))
                return False

        return (True, None) if certificate else True

    def frank_network(self):
        r"""
        Return Frank's network of the poset.

        This is defined in Section 8 of [BF1999]_.

        OUTPUT:

        A pair `(G, e)`, where `G` is Frank's network of `P` encoded as a
        :class:`DiGraph`, and `e` is the cost function on its edges encoded
        as a dictionary (indexed by these edges, which in turn are encoded
        as tuples of 2 vertices).

        .. NOTE::

            Frank's network of `P` is a certain directed graph with `2|P| + 2`
            vertices, defined in Section 8 of [BF1999]_. Its set of vertices
            consists of two vertices `(0, p)` and `(1, p)` for each element
            `p` of `P`, as well as two vertices `(-1, 0)` and `(2, 0)`.
            (These notations are not the ones used in [BF1999]_; see the table
            below for their relation.) The edges are:

            - for each `p` in `P`, an edge from `(-1, 0)` to `(0, p)`;

            - for each `p` in `P`, an edge from `(1, p)` to `(2, 0)`;

            - for each `p` and `q` in `P` such that `p \geq q`, an edge from
              `(0, p)` to `(1, q)`.

            We make this digraph into a network in the sense of flow theory as
            follows: The vertex `(-1, 0)` is considered as the source of this
            network, and the vertex `(2, 0)` as the sink. The cost function is
            defined to be `1` on the edge from `(0, p)` to `(1, p)` for each
            `p \in P`, and to be `0` on every other edge. The capacity is `1`
            on each edge. Here is how to translate this notations into that
            used in [BF1999]_::

              our notations                    [BF1999]
                 (-1, 0)                          s
                 (0, p)                          x_p
                 (1, p)                          y_p
                 (2, 0)                           t
                  a[e]                           a(e)

        EXAMPLES::

            sage: ps = [[16,12,14,-13],[[12,14],[14,-13],[12,16],[16,-13]]]
            sage: G, e = Poset(ps).frank_network()
            sage: G.edges(sort=True)
            [((-1, 0), (0, -13), None), ((-1, 0), (0, 12), None), ((-1, 0), (0, 14), None), ((-1, 0), (0, 16), None), ((0, -13), (1, -13), None), ((0, -13), (1, 12), None), ((0, -13), (1, 14), None), ((0, -13), (1, 16), None), ((0, 12), (1, 12), None), ((0, 14), (1, 12), None), ((0, 14), (1, 14), None), ((0, 16), (1, 12), None), ((0, 16), (1, 16), None), ((1, -13), (2, 0), None), ((1, 12), (2, 0), None), ((1, 14), (2, 0), None), ((1, 16), (2, 0), None)]
            sage: e
            {((-1, 0), (0, -13)): 0,
             ((-1, 0), (0, 12)): 0,
             ((-1, 0), (0, 14)): 0,
             ((-1, 0), (0, 16)): 0,
             ((0, -13), (1, -13)): 1,
             ((0, -13), (1, 12)): 0,
             ((0, -13), (1, 14)): 0,
             ((0, -13), (1, 16)): 0,
             ((0, 12), (1, 12)): 1,
             ((0, 14), (1, 12)): 0,
             ((0, 14), (1, 14)): 1,
             ((0, 16), (1, 12)): 0,
             ((0, 16), (1, 16)): 1,
             ((1, -13), (2, 0)): 0,
             ((1, 12), (2, 0)): 0,
             ((1, 14), (2, 0)): 0,
             ((1, 16), (2, 0)): 0}
            sage: qs = [[1,2,3,4,5,6,7,8,9],[[1,3],[3,4],[5,7],[1,9],[2,3]]]
            sage: Poset(qs).frank_network()
            (Digraph on 20 vertices,
             {((-1, 0), (0, 1)): 0,
              ((-1, 0), (0, 2)): 0,
              ((-1, 0), (0, 3)): 0,
              ((-1, 0), (0, 4)): 0,
              ((-1, 0), (0, 5)): 0,
              ((-1, 0), (0, 6)): 0,
              ((-1, 0), (0, 7)): 0,
              ((-1, 0), (0, 8)): 0,
              ((-1, 0), (0, 9)): 0,
              ((0, 1), (1, 1)): 1,
              ((0, 2), (1, 2)): 1,
              ((0, 3), (1, 1)): 0,
              ((0, 3), (1, 2)): 0,
              ((0, 3), (1, 3)): 1,
              ((0, 4), (1, 1)): 0,
              ((0, 4), (1, 2)): 0,
              ((0, 4), (1, 3)): 0,
              ((0, 4), (1, 4)): 1,
              ((0, 5), (1, 5)): 1,
              ((0, 6), (1, 6)): 1,
              ((0, 7), (1, 5)): 0,
              ((0, 7), (1, 7)): 1,
              ((0, 8), (1, 8)): 1,
              ((0, 9), (1, 1)): 0,
              ((0, 9), (1, 9)): 1,
              ((1, 1), (2, 0)): 0,
              ((1, 2), (2, 0)): 0,
              ((1, 3), (2, 0)): 0,
              ((1, 4), (2, 0)): 0,
              ((1, 5), (2, 0)): 0,
              ((1, 6), (2, 0)): 0,
              ((1, 7), (2, 0)): 0,
              ((1, 8), (2, 0)): 0,
              ((1, 9), (2, 0)): 0})

        AUTHOR:

        - Darij Grinberg (2013-05-09)
        """
        from sage.graphs.digraph import DiGraph
        P0 = [(0, i) for i in self]
        pdict = {(-1, 0): P0, (2, 0): []}
        for i in self:
            pdict[(0, i)] = [(1, j) for j in self if self.ge(i, j)]
            pdict[(1, i)] = [(2, 0)]
        G = DiGraph(pdict, format="dict_of_lists")
        a = {(u, v): 0 for (u, v, l) in G.edge_iterator()}
        for i in self:
            a[((0, i), (1, i))] = 1
        return (G, a)

    @combinatorial_map(name="Greene-Kleitman partition")
    def greene_shape(self):
        r"""
        Return the Greene-Kleitman partition of ``self``.

        The Greene-Kleitman partition of a finite poset `P` is the partition
        `(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)`, where `c_k` is the
        maximum cardinality of a union of `k` chains of `P`. Equivalently,
        this is the conjugate of the partition `(a_1 - a_0, a_2 - a_1, a_3 -
        a_2, \ldots)`, where `a_k` is the maximum cardinality of a union of
        `k` antichains of `P`.

        See many sources, e. g., [BF1999]_, for proofs of this equivalence.

        EXAMPLES::

            sage: P = Poset([[3,2,1],[[3,1],[2,1]]])
            sage: P.greene_shape()
            [2, 1]
            sage: P = Poset([[1,2,3,4],[[1,4],[2,4],[4,3]]])
            sage: P.greene_shape()
            [3, 1]
            sage: P = Poset([[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[[1,4],[2,4],[4,3]]])
            sage: P.greene_shape()
            [3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
            sage: P = Poset([[],[]])
            sage: P.greene_shape()
            []

        AUTHOR:

        - Darij Grinberg (2013-05-09)
        """
        from sage.combinat.partition import Partition
        (G, a) = self.frank_network()
        n = len(self)
        chron = _ford_fulkerson_chronicle(G, (-1, 0), (2, 0), a)
        size = 0
        ps = []
        part = 0
        (pold, vold) = (0, 0)
        while size != n:
            (p, v) = next(chron)
            if v > vold:
                size += p
                if part > 0:
                    ps.append(part)
            elif p > pold:
                part += 1
            (pold, vold) = (p, v)
        ps.reverse()
        return Partition(ps)

    def p_partition_enumerator(self, tup, R, weights=None, check=False):
        r"""
        Return a `P`-partition enumerator of ``self``.

        Given a total order `\prec` on the elements of a finite poset `P`
        (the order of `P` and the total order `\prec` can be unrelated; in
        particular, the latter does not have to extend the former), a
        `P`-partition enumerator is the quasisymmetric function
        `\sum_f \prod_{p \in P} x_{f(p)}`, where the first sum is taken over
        all `P`-partitions `f`.

        A `P`-partition is a function `f : P \to \{1,2,3,...\}` satisfying
        the following properties for any two elements `i` and `j` of `P`
        satisfying `i <_P j`:

        - if `i \prec j` then `f(i) \leq f(j)`,

        - if `j \prec i` then `f(i) < f(j)`.

        The optional argument ``weights`` allows constructing a
        generalized ("weighted") version of the `P`-partition enumerator.
        Namely, ``weights`` should be a dictionary whose keys are the
        elements of ``P``.
        Then, the generalized `P`-partition enumerator corresponding to
        weights ``weights`` is `\sum_f \prod_{p \in P} x_{f(p)}^{w(p)}`,
        where the sum is again over all `P`-partitions `f`. Here,
        `w(p)` is ``weights[p]``. The classical `P`-partition enumerator
        is the particular case obtained when all `p` satisfy `w(p) = 1`.

        In the language of [Grinb2016a]_, the generalized `P`-partition
        enumerator is the quasisymmetric function
        `\Gamma\left(\mathbf{E}, w\right)`, where `\mathbf{E}` is the
        special double poset `(P, <_P, \prec)`, and where
        `w` is the dictionary ``weights`` (regarded as a function from
        `P` to the positive integers).

        INPUT:

        - ``tup`` -- the tuple containing all elements of `P` (each of
          them exactly once), in the order dictated by the total order
          `\prec`

        - ``R`` -- a commutative ring

        - ``weights`` -- (optional) a dictionary of positive integers,
          indexed by elements of `P`; any missing item will be understood
          as `1`

        OUTPUT:

        The `P`-partition enumerator of ``self`` according to ``tup`` in the
        algebra `QSym` of quasisymmetric functions over the base ring `R`.

        EXAMPLES::

            sage: P = Poset([[1,2,3,4],[[1,4],[2,4],[4,3]]])
            sage: FP = P.p_partition_enumerator((3,1,2,4), QQ, check=True); FP
            2*M[1, 1, 1, 1] + 2*M[1, 2, 1] + M[2, 1, 1] + M[3, 1]

            sage: expansion = FP.expand(5)
            sage: xs = expansion.parent().gens()
            sage: expansion == sum([xs[a]*xs[b]*xs[c]*xs[d] for a in range(5) for b in range(5) for c in range(5) for d in range(5) if a <= b and c <= b and b < d])
            True

            sage: P = Poset([[],[]])
            sage: FP = P.p_partition_enumerator((), QQ, check=True); FP
            M[]

        With the ``weights`` parameter::

            sage: P = Poset([[1,2,3,4],[[1,4],[2,4],[4,3]]])
            sage: FP = P.p_partition_enumerator((3,1,2,4), QQ, weights={1: 1, 2: 2, 3: 1, 4: 1}, check=True); FP
            M[1, 2, 1, 1] + M[1, 3, 1] + M[2, 1, 1, 1] + M[2, 2, 1] + M[3, 1, 1] + M[4, 1]
            sage: FP = P.p_partition_enumerator((3,1,2,4), QQ, weights={2: 2}, check=True); FP
            M[1, 2, 1, 1] + M[1, 3, 1] + M[2, 1, 1, 1] + M[2, 2, 1] + M[3, 1, 1] + M[4, 1]

            sage: P = Poset([['a','b','c'], [['a','b'], ['a','c']]])
            sage: FP = P.p_partition_enumerator(('b','c','a'), QQ, weights={'a': 3, 'b': 5, 'c': 7}, check=True); FP
            M[3, 5, 7] + M[3, 7, 5] + M[3, 12]

            sage: P = Poset([['a','b','c'], [['a','c'], ['b','c']]])
            sage: FP = P.p_partition_enumerator(('b','c','a'), QQ, weights={'a': 3, 'b': 5, 'c': 7}, check=True); FP
            M[3, 5, 7] + M[3, 12] + M[5, 3, 7] + M[8, 7]
            sage: FP = P.p_partition_enumerator(('a','b','c'), QQ, weights={'a': 3, 'b': 5, 'c': 7}, check=True); FP
            M[3, 5, 7] + M[3, 12] + M[5, 3, 7] + M[5, 10] + M[8, 7] + M[15]
        """
        if check:
            if sorted(self.list()) != sorted(tup):
                raise ValueError("the elements of tup are not those of P")
        from sage.combinat.composition import Composition
        from sage.combinat.ncsf_qsym.qsym import QuasiSymmetricFunctions
        QR = QuasiSymmetricFunctions(R)
        n = len(tup)
        res = QR.zero()
        tupdict = dict(zip(tup, range(n)))
        if weights is None:
            # The simple case: ``weights == None``.
            F = QR.Fundamental()
            for lin in self.linear_extensions(facade=True):
                descents = [i + 1 for i in range(n-1) if tupdict[lin[i]] > tupdict[lin[i+1]]]
                res += F(Composition(from_subset=(descents, n)))
            return res
        for lin in self.linear_extensions(facade=True):
            M = QR.Monomial()
            lin_weights = Composition([weights.get(lin[i], 1) for i in range(n)])
            descents = [i + 1 for i in range(n-1) if tupdict[lin[i]] > tupdict[lin[i+1]]]
            d_c = Composition(from_subset=(descents, n))
            for comp in d_c.finer():
                res += M[lin_weights.fatten(comp)]
        return res

    def cuts(self):
        r"""
        Return the list of cuts of the poset ``self``.

        A cut is a subset `A` of ``self`` such that the set of lower
        bounds of the set of upper bounds of `A` is exactly `A`.

        The cuts are computed here using the maximal independent sets in the
        auxiliary graph defined as `P \times [0,1]` with an edge
        from `(x, 0)` to `(y, 1)` if
        and only if `x \not\geq_P y`. See the end of section 4 in [JRJ94]_.

        EXAMPLES::

            sage: P = posets.AntichainPoset(3)
            sage: Pc = P.cuts()
            sage: Pc # random
            [frozenset({0}),
             frozenset(),
             frozenset({0, 1, 2}),
             frozenset({2}),
             frozenset({1})]
            sage: sorted(list(c) for c in Pc)
            [[], [0], [0, 1, 2], [1], [2]]

        .. SEEALSO::

            :meth:`completion_by_cuts`
        """
        from sage.graphs.graph import Graph
        from sage.graphs.independent_sets import IndependentSets
        auxg = Graph({(u, 0): [(v, 1) for v in self if not self.ge(u, v)]
                      for u in self}, format="dict_of_lists")
        auxg.add_vertices([(v, 1) for v in self])
        return [frozenset([xa for xa, xb in c if xb == 0])
                for c in IndependentSets(auxg, maximal=True)]

    def completion_by_cuts(self):
        """
        Return the completion by cuts of ``self``.

        This is the smallest lattice containing the poset. This is also
        called the Dedekind-MacNeille completion.

        See the :wikipedia:`Dedekind-MacNeille completion`.

        OUTPUT:

        - a finite lattice

        EXAMPLES::

            sage: P = posets.PentagonPoset()
            sage: P.completion_by_cuts().is_isomorphic(P)
            True

            sage: Y = Poset({1: [2], 2: [3, 4]})
            sage: trafficsign = LatticePoset({1: [2], 2: [3, 4], 3: [5], 4: [5]})
            sage: L = Y.completion_by_cuts()
            sage: L.is_isomorphic(trafficsign)
            True

            sage: P = posets.SymmetricGroupBruhatOrderPoset(3)
            sage: Q = P.completion_by_cuts(); Q
            Finite lattice containing 7 elements

        .. SEEALSO::

            :meth:`cuts`,
            :meth:`~sage.categories.finite_lattice_posets.FiniteLatticePosets.ParentMethods.irreducibles_poset`

        TESTS::

            sage: Poset().completion_by_cuts()
            Finite lattice containing 0 elements
        """
        from sage.combinat.posets.lattices import LatticePoset
        if self.cardinality() == 0:
            return LatticePoset({})
        return LatticePoset((self.cuts(), lambda a, b: a.issuperset(b)))

    def incidence_algebra(self, R, prefix='I'):
        r"""
        Return the incidence algebra of ``self`` over ``R``.

        OUTPUT:

        An instance of :class:`sage.combinat.posets.incidence_algebras.IncidenceAlgebra`.

        EXAMPLES::

            sage: P = posets.BooleanLattice(4)
            sage: P.incidence_algebra(QQ)
            Incidence algebra of Finite lattice containing 16 elements
             over Rational Field
        """
        from sage.combinat.posets.incidence_algebras import IncidenceAlgebra
        return IncidenceAlgebra(R, self, prefix)

    @cached_method(key=lambda self, x, y, l: (x, y))
    def _kl_poly(self, x=None, y=None, canonical_labels=None):
        r"""
        Cached Kazhdan-Lusztig polynomial of ``self`` for generic `q`.

        EXAMPLES::

            sage: L = posets.SymmetricGroupWeakOrderPoset(4)
            sage: L._kl_poly()
            1
            sage: x = '2314'
            sage: y = '3421'
            sage: L._kl_poly(x, y)
            -q + 1

        .. SEEALSO::

            :meth:`kazhdan_lusztig_polynomial`

        AUTHORS:

        - Travis Scrimshaw (27-12-2014)
        """
        R = PolynomialRing(ZZ, 'q')
        q = R.gen(0)

        # Handle some special cases
        if self.cardinality() == 0:
            return q.parent().zero()
        if not self.rank():
            return q.parent().one()

        if canonical_labels is None:
            canonical_labels = x is None and y is None

        if x is not None or y is not None:
            if x == y:
                return q.parent().one()
            if x is None:
                x = self.minimal_elements()[0]
            if y is None:
                y = self.maximal_elements()[0]
            if not self.le(x, y):
                return q.parent().zero()
            P = self.subposet(self.interval(x, y))
            return P.kazhdan_lusztig_polynomial(q=q, canonical_labels=canonical_labels)

        min_elt = self.minimal_elements()[0]
        if canonical_labels:
            def sublat(P):
                return self.subposet(P).canonical_label()
        else:
            def sublat(P):
                return self.subposet(P)
        poly = -sum(sublat(self.order_ideal([x])).characteristic_polynomial() *
                    sublat(self.order_filter([x])).kazhdan_lusztig_polynomial()
                    for x in self if x != min_elt)
        tr = floor(self.rank()/2) + 1
        ret = poly.truncate(tr)
        return ret(q=q)

    def kazhdan_lusztig_polynomial(self, x=None, y=None, q=None, canonical_labels=None):
        r"""
        Return the Kazhdan-Lusztig polynomial `P_{x,y}(q)` of the poset.

        The poset is expected to be ranked.

        We follow the definition given in [EPW14]_. Let `G` denote a
        graded poset with unique minimal and maximal elements and `\chi_G`
        denote the characteristic polynomial of `G`. Let `I_x` and `F^x`
        denote the principal order ideal and filter of `x` respectively.
        Define the *Kazhdan-Lusztig polynomial* of `G` as the unique
        polynomial `P_G(q)` satisfying the following:

        1. If `\operatorname{rank} G = 0`, then `P_G(q) = 1`.
        2. If `\operatorname{rank} G > 0`, then `\deg P_G(q) <
           \frac{1}{2} \operatorname{rank} G`.
        3. We have

           .. MATH::

                q^{\operatorname{rank} G} P_G(q^{-1})
                = \sum_{x \in G} \chi_{I_x}(q) P_{F^x}(q).

        We then extend this to `P_{x,y}(q)` by considering the subposet
        corresponding to the (closed) interval `[x, y]`. We also
        define `P_{\emptyset}(q) = 0` (so if `x \not\leq y`,
        then `P_{x,y}(q) = 0`).

        INPUT:

        - ``q`` -- (default: `q \in \ZZ[q]`) the indeterminate `q`
        - ``x`` -- (default: the minimal element) the element `x`
        - ``y`` -- (default: the maximal element) the element `y`
        - ``canonical_labels`` -- (optional) for subposets, use the
          canonical labeling (this can limit recursive calls for posets
          with large amounts of symmetry, but producing the labeling
          takes time); if not specified, this is ``True`` if ``x``
          and ``y`` are both not specified and ``False`` otherwise

        EXAMPLES::

            sage: L = posets.BooleanLattice(3)
            sage: L.kazhdan_lusztig_polynomial()
            1

        ::

            sage: L = posets.SymmetricGroupWeakOrderPoset(4)
            sage: L.kazhdan_lusztig_polynomial()
            1
            sage: x = '2314'
            sage: y = '3421'
            sage: L.kazhdan_lusztig_polynomial(x, y)
            -q + 1
            sage: L.kazhdan_lusztig_polynomial(x, y, var('t'))
            -t + 1

        AUTHORS:

        - Travis Scrimshaw (27-12-2014)
        """
        if not self.is_ranked():
            raise ValueError("the poset is not ranked")
        if q is None:
            q = PolynomialRing(ZZ, 'q').gen(0)
        poly = self._kl_poly(x, y, canonical_labels)
        return poly(q=q)

    def is_induced_subposet(self, other):
        r"""
        Return ``True`` if the poset is an induced subposet of ``other``, and
        ``False`` otherwise.

        A poset `P` is an induced subposet of `Q` if every element
        of `P` is an element of `Q`, and `x \le_P y` iff `x \le_Q y`.
        Note that "induced" here has somewhat different meaning compared
        to that of graphs.

        INPUT:

        - ``other``, a poset.

        .. NOTE::

            This method does not check whether the poset is a
            *isomorphic* (i.e., up to relabeling) subposet of ``other``,
            but only if ``other`` directly contains the poset as an
            induced subposet. For isomorphic subposets see
            :meth:`has_isomorphic_subposet`.

        EXAMPLES::

            sage: P = Poset({1:[2, 3]})
            sage: Q = Poset({1:[2, 4], 2:[3]})
            sage: P.is_induced_subposet(Q)
            False
            sage: R = Poset({0:[1], 1:[3, 4], 3:[5], 4:[2]})
            sage: P.is_induced_subposet(R)
            True

        TESTS::

            sage: P = Poset({2:[1]})
            sage: Poset().is_induced_subposet(P)
            True
            sage: Poset().is_induced_subposet(Poset())
            True
            sage: P.is_induced_subposet(Poset())
            False

        Bad input::

            sage: Poset().is_induced_subposet('junk')
            Traceback (most recent call last):
            ...
            AttributeError: 'str' object has no attribute 'subposet'
        """
        if (not self._is_facade or (isinstance(other, FinitePoset) and
                                    not other._is_facade)):
            raise TypeError("the function is not defined on non-facade posets")
        # TODO: When we have decided if
        # Poset({'x':[42]}) == LatticePoset({'x':[42]})
        # or not, either remove this note or remove .hasse_diagram() below.
        return (set(self).issubset(set(other)) and
                other.subposet(self).hasse_diagram() == self.hasse_diagram())

    def _libgap_(self):
        """
        Conversion to gap.

        This uses the QPA package (https://folk.ntnu.no/oyvinso/QPA/).

        EXAMPLES::

            sage: P = posets.TamariLattice(3)
            sage: libgap(P)  # optional - gap_packages
            <A poset on 5 points>
            sage: A = libgap(GF(2)).PosetAlgebra(P); A  # optional - gap_packages
            <GF(2)[<quiver with 5 vertices and 5 arrows>]/<two-sided ideal in <GF(2)[<quiver with 5 vertices and 5 arrows>]>, (1 generators)>>
            sage: A.Dimension()  # optional - gap_packages
            13
        """
        from sage.libs.gap.libgap import libgap
        libgap.LoadPackage("QPA")
        L = list(self)
        g = libgap.Poset(L, [self.principal_order_filter(x) for x in L])
        return g

    def _macaulay2_init_(self, macaulay2=None):
        """
        Conversion to Macaulay2.

        This uses the ``Posets`` package.

        EXAMPLES::

            sage: P = posets.PentagonPoset()
            sage: P._macaulay2_init_()
            'needsPackage "Posets";poset({0,1,2,3,4},{{0,1},{0,2},{1,4},{2,3},{3,4}})'

            sage: P = Poset({1:[2],2:[]})
            sage: macaulay2('needsPackage "Posets"')   # optional - macaulay2
            Posets
            sage: macaulay2(P)   # optional - macaulay2
            Relation Matrix: | 1 1 |
                             | 0 1 |
        """
        H = self._hasse_diagram
        txt = 'needsPackage "Posets";'
        txt += "poset({%s},{" % ','.join(str(x) for x in H)
        txt += ",".join("{%s,%s}" % (str(x), str(y))
                        for x, y in H.cover_relations_iterator())
        return txt + "})"


FinitePoset._dual_class = FinitePoset

# ------- Posets -------


class FinitePosets_n(UniqueRepresentation, Parent):
    r"""
    The finite enumerated set of all posets on `n` elements, up to an isomorphism.

    EXAMPLES::

        sage: P = Posets(3)
        sage: P.cardinality()
        5
        sage: for p in P: print(p.cover_relations())
        []
        [[1, 2]]
        [[0, 1], [0, 2]]
        [[0, 1], [1, 2]]
        [[1, 2], [0, 2]]
    """

    def __init__(self, n):
        r"""
        EXAMPLES::

            sage: P = Posets(3); P
            Posets containing 3 elements
            sage: P.category()
            Category of finite enumerated sets
            sage: P.__class__
            <class 'sage.combinat.posets.posets.FinitePosets_n_with_category'>
            sage: TestSuite(P).run()
        """
        Parent.__init__(self, category=FiniteEnumeratedSets())
        self._n = n

    def _repr_(self):
        r"""
        EXAMPLES::

            sage: P = Posets(3)
            sage: P._repr_()
            'Posets containing 3 elements'
        """
        return "Posets containing %s elements" % self._n

    def __contains__(self, P):
        """
        EXAMPLES::

            sage: posets.PentagonPoset() in Posets(5)
            True
            sage: posets.PentagonPoset() in Posets(3)
            False
            sage: 1 in Posets(3)
            False
        """
        return P in FinitePosets() and P.cardinality() == self._n

    def __iter__(self):
        """
        Return an iterator of representatives of the isomorphism classes
        of finite posets of a given size.

        .. NOTE::

            This uses the DiGraph iterator as a backend to construct
            transitively-reduced, acyclic digraphs.

        EXAMPLES::

            sage: P = Posets(2)
            sage: list(P)
            [Finite poset containing 2 elements, Finite poset containing 2 elements]
        """
        from sage.graphs.digraph_generators import DiGraphGenerators
        for dig in DiGraphGenerators()(self._n, is_poset):
            # We need to relabel the digraph since range(self._n) must be a linear
            # extension. Too bad we need to compute this again. TODO: Fix this.
            label_dict = dict(zip(dig.topological_sort(), range(dig.order())))
            yield FinitePoset(dig.relabel(label_dict, inplace=False))

    def cardinality(self, from_iterator=False):
        r"""
        Return the cardinality of this object.

        .. note::

            By default, this returns pre-computed values obtained from
            the On-Line Encyclopedia of Integer Sequences (:oeis:`A000112`).
            To override this, pass the argument ``from_iterator=True``.

        EXAMPLES::

            sage: P = Posets(3)
            sage: P.cardinality()
            5
            sage: P.cardinality(from_iterator=True)
            5
        """
        # Obtained from The On-Line Encyclopedia of Integer Sequences;
        # this is sequence number A000112.
        known_values = [1, 1, 2, 5, 16, 63, 318, 2045, 16999, 183231,
                2567284, 46749427, 1104891746, 33823827452, 1338193159771,
                68275077901156, 4483130665195087]
        if not from_iterator and self._n < len(known_values):
            return Integer(known_values[self._n])
        else:
            return super(FinitePosets_n, self).cardinality()


# For backward compatibility of pickles of the former Posets()
Posets_all = Posets

# ------- Miscellaneous functions -------


def is_poset(dig):
    r"""
    Return ``True`` if a directed graph is acyclic and transitively
    reduced, and ``False`` otherwise.

    EXAMPLES::

        sage: from sage.combinat.posets.posets import is_poset
        sage: dig = DiGraph({0:[2, 3], 1:[3, 4, 5], 2:[5], 3:[5], 4:[5]})
        sage: is_poset(dig)
        False
        sage: is_poset(dig.transitive_reduction())
        True
    """
    return dig.is_directed_acyclic() and dig.is_transitively_reduced()


def _ford_fulkerson_chronicle(G, s, t, a):
    r"""
    Iterate through the Ford-Fulkerson algorithm for an acyclic directed
    graph with all edge capacities equal to `1`. This is an auxiliary algorithm
    for use by the :meth:`FinitePoset.greene_shape` method of finite posets,
    and is lacking some of the functionality that a general Ford-Fulkerson
    algorithm implementation should have.

    INPUT:

    - ``G`` -- an acyclic directed graph

    - ``s`` -- a vertex of `G` as the source

    - ``t`` -- a vertex of `G` as the sink

    - ``a`` -- a cost function (on the set of edges of ``G``) encoded as
      a dictionary. The keys of this dictionary are encoded as pairs
      of vertices.

    OUTPUT:

    An iterator which iterates through the values `(p, v)` during the
    application of the Ford-Fulkerson algorithm applied to the graph
    `G` with source `s`, sink `t`, cost function `a` and capacity `1`
    on each edge. Here, `p` denotes the value of the potential, and `v`
    denotes the value of the flow at every moment during the execution
    of the algorithm. The algorithm starts at `(p, v) = (0, 0)`.
    Every time ``next()`` is called, the iterator performs one step of
    the algorithm (incrementing either `p` or `v`) and yields the
    resulting pair `(p, v)`. Note that `(0, 0)` is never yielded.
    The iterator goes on for eternity, since the stopping condition
    is not implemented. This is OK for use in the ``greene_partition``
    function, since that one knows when to stop.

    The notation used here is that of Section 7 of [BF1999]_.

    .. WARNING::

        This method is tailor-made for its use in the
        :meth:`FinitePoset.greene_shape()` method of a finite poset. It is not
        very useful in general. First of all, as said above, the iterator
        does not know when to halt. Second, `G` needs to be acyclic for it
        to correctly work. This must be amended if this method is ever to be
        used outside the Greene-Kleitman partition construction. For the
        Greene-Kleitman partition, this is a non-issue since Frank's network
        is always acyclic.

    EXAMPLES::

        sage: from sage.combinat.posets.posets import _ford_fulkerson_chronicle
        sage: G = DiGraph({1: [3,6,7], 2: [4], 3: [7], 4: [], 6: [7,8], 7: [9], 8: [9,12], 9: [], 10: [], 12: []})
        sage: s = 1
        sage: t = 9
        sage: (1, 6, None) in G.edges()
        True
        sage: (1, 6) in G.edges()
        False
        sage: a = {(1, 6): 4, (2, 4): 0, (1, 3): 4, (1, 7): 1, (3, 7): 6, (7, 9): 1, (6, 7): 3, (6, 8): 1, (8, 9): 0, (8, 12): 2}
        sage: ffc = _ford_fulkerson_chronicle(G, s, t, a)
        sage: next(ffc)
        (1, 0)
        sage: next(ffc)
        (2, 0)
        sage: next(ffc)
        (2, 1)
        sage: next(ffc)
        (3, 1)
        sage: next(ffc)
        (4, 1)
        sage: next(ffc)
        (5, 1)
        sage: next(ffc)
        (5, 2)
        sage: next(ffc)
        (6, 2)
        sage: next(ffc)
        (7, 2)
        sage: next(ffc)
        (8, 2)
        sage: next(ffc)
        (9, 2)
        sage: next(ffc)
        (10, 2)
        sage: next(ffc)
        (11, 2)
    """
    from sage.graphs.digraph import DiGraph

    # pi: potential function as a dictionary.
    pi = {v: 0 for v in G.vertex_iterator()}
    # p: value of the potential pi.
    p = 0

    # f: flow function as a dictionary.
    f = {(u, v): 0 for (u, v, l) in G.edge_iterator()}
    # val: value of the flow f. (Cannot call it v due to Python's asinine
    # handling of for loops.)
    val = 0

    # capacity: capacity function as a dictionary. Here, just the
    # indicator function of the set of arcs of G.
    capacity = {(u, v): 1 for (u, v, l) in G.edge_iterator()}

    while True:

        # Step MC1 in Britz-Fomin, Algorithm 7.2.

        # Gprime: directed graph G' from Britz-Fomin, Section 7.
        Gprime = DiGraph()
        Gprime.add_vertices(G.vertices())
        for (u, v, l) in G.edge_iterator():
            if pi[v] - pi[u] == a[(u, v)]:
                if f[(u, v)] < capacity[(u, v)]:
                    Gprime.add_edge(u, v)
                elif f[(u, v)] > 0:
                    Gprime.add_edge(v, u)

        # X: list of vertices of G' reachable from s, along with
        # the shortest paths from s to them.
        X = Gprime.shortest_paths(s)
        if t in X:
            # Step MC2a in Britz-Fomin, Algorithm 7.2.
            shortest_path = X[t]
            shortest_path_in_edges = zip(shortest_path[:-1], shortest_path[1:])
            for (u, v) in shortest_path_in_edges:
                if v in G.neighbors_out(u):
                    f[(u, v)] += 1
                else:
                    f[(v, u)] -= 1
            val += 1
        else:
            # Step MC2b in Britz-Fomin, Algorithm 7.2.
            for v in G.vertex_iterator():
                if v not in X:
                    pi[v] += 1
            p += 1

        yield (p, val)
