Hasse diagrams of posets

class sage.combinat.posets.hasse_diagram.HasseDiagram(data=None, pos=None, loops=None, format=None, boundary=None, weighted=None, implementation='c_graph', data_structure='sparse', vertex_labels=True, name=None, multiedges=None, convert_empty_dict_labels_to_None=None, sparse=True, immutable=False)

Bases: sage.graphs.digraph.DiGraph

The Hasse diagram of a poset. This is just a transitively-reduced, directed, acyclic graph without loops or multiple edges.

Note

We assume that range(n) is a linear extension of the poset. That is, range(n) is the vertex set and a topological sort of the digraph.

This should not be called directly, use Poset instead; all type checking happens there.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,2],1:[3],2:[3],3:[]}); H
Hasse diagram of a poset containing 4 elements
sage: TestSuite(H).run()
antichains(element_class=<type 'list'>)

Returns all antichains of self, organized as a prefix tree

INPUT:

  • element_class – (default:list) an iterable type

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: H = P._hasse_diagram
sage: A = H.antichains()
sage: list(A)
[[], [0], [1], [1, 2], [1, 3], [2], [3], [4]]
sage: A.cardinality()
8
sage: [1,3] in A
True
sage: [1,4] in A
False

TESTS:

sage: TestSuite(A).run(skip = "_test_pickling")

Note

It’s actually the pickling of the cached method coxeter_transformation() that fails ...

TESTS:

sage: A = Poset()._hasse_diagram.antichains()
sage: list(A)
[[]]
sage: TestSuite(A).run()
antichains_iterator()

Return an iterator over the antichains of the poset.

Note

The algorithm is based on Freese-Jezek-Nation p. 226. It does a depth first search through the set of all antichains organized in a prefix tree.

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: H = P._hasse_diagram
sage: H.antichains_iterator()
<generator object antichains_iterator at ...>
sage: list(H.antichains_iterator())
[[], [4], [3], [2], [1], [1, 3], [1, 2], [0]]

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,2],1:[4],2:[3],3:[4]})
sage: list(H.antichains_iterator())
[[], [4], [3], [2], [1], [1, 3], [1, 2], [0]]

sage: H = HasseDiagram({0:[],1:[],2:[]})
sage: list(H.antichains_iterator())
[[], [2], [1], [1, 2], [0], [0, 2], [0, 1], [0, 1, 2]]

sage: H = HasseDiagram({0:[1],1:[2],2:[3],3:[4]})
sage: list(H.antichains_iterator())
[[], [4], [3], [2], [1], [0]]

TESTS:

sage: H = Poset()._hasse_diagram
sage: list(H.antichains_iterator())
[[]]
are_comparable(i, j)

Returns whether i and j are comparable in the poset

INPUT:

  • i, j – vertices of this Hasse diagram

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: H = P._hasse_diagram
sage: H.are_comparable(1,2)
False
sage: [ (i,j) for i in H.vertices() for j in H.vertices() if H.are_comparable(i,j)]
[(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (1, 0), (1, 1), (1, 4), (2, 0), (2, 2), (2, 3), (2, 4), (3, 0), (3, 2), (3, 3), (3, 4), (4, 0), (4, 1), (4, 2), (4, 3), (4, 4)]
are_incomparable(i, j)

Returns whether i and j are incomparable in the poset

INPUT:

  • i, j – vertices of this Hasse diagram

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: H = P._hasse_diagram
sage: H.are_incomparable(1,2)
True
sage: [ (i,j) for i in H.vertices() for j in H.vertices() if H.are_incomparable(i,j)]
[(1, 2), (1, 3), (2, 1), (3, 1)]
bottom()

Returns the bottom 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
cardinality()

Returns the number of elements in the poset.

EXAMPLES:

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

TESTS:

For a time, this function was named size(), which would override the same-named method of the underlying digraph. Trac #8735 renamed this method to cardinality() with a deprecation warning. Trac #11214 removed the warning since code for graphs was raising the warning inadvertently. This tests that size() for a Hasse diagram returns the number of edges in the digraph.

sage: L = Posets.BooleanLattice(5)
sage: H = L.hasse_diagram()
sage: H.size()
80
sage: H.size() == H.num_edges()
True
chains(element_class=<type 'list'>, exclude=None)

Return all chains of self, organized as a prefix tree.

INPUT:

  • element_class – (default: list) an iterable type
  • exclude – elements of the poset to be excluded (default: None)

OUTPUT:

The enumerated set (with a forest structure given by prefix ordering) consisting of all chains of self, each of which is given as an element_class.

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: H = P._hasse_diagram
sage: A = H.chains()
sage: list(A)
[[], [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]]
sage: A.cardinality()
20
sage: [1,3] in A
False
sage: [1,4] in A
True

One can exclude some vertices:

sage: list(H.chains(exclude=[4, 3]))
[[], [0], [0, 1], [0, 2], [1], [2]]

The element_class keyword determines how the chains are being returned:

sage: P = Poset({1: [2, 3], 2: [4]}) sage: list(P._hasse_diagram.chains(element_class=tuple)) [(), (0,), (0, 1), (0, 1, 2), (0, 2), (0, 3), (1,), (1, 2), (2,), (3,)] sage: list(P._hasse_diagram.chains()) [[], [0], [0, 1], [0, 1, 2], [0, 2], [0, 3], [1], [1, 2], [2], [3]]

(Note that taking the Hasse diagram has renamed the vertices.)

sage: list(P._hasse_diagram.chains(element_class=tuple, exclude=[0])) [(), (1,), (1, 2), (2,), (3,)]

See also

antichains()

closed_interval(x, y)

Return a list of the elements \(z\) of self such that \(x \leq z \leq y\). The order is that induced by the ordering in self.linear_extension.

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: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram(dag)
sage: I = set([2,5,6,4,7])
sage: I == set(H.interval(2,7))
True
complements()

Return a list l such that l[i] is a complement of i in self, or None if no such complement exists.

A complement of x is an element y such that the meet of x and y is the bottom element of self and the join of x and y is the top element of self.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,2,3],1:[4],2:[4],3:[4]})
sage: H.complements()
[4, 3, 3, 2, 0]

sage: H = HasseDiagram({0:[1,2],1:[3],2:[3],3:[4]})
sage: H.complements()
[4, None, None, None, 0]
cover_relations(element=None)

TESTS:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[2,3], 1:[3,4], 2:[5], 3:[5], 4:[5]})
sage: H.cover_relations()
[(0, 2), (0, 3), (1, 3), (1, 4), (2, 5), (3, 5), (4, 5)]
cover_relations_iterator()

TESTS:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[2,3], 1:[3,4], 2:[5], 3:[5], 4:[5]})
sage: list(H.cover_relations_iterator())
[(0, 2), (0, 3), (1, 3), (1, 4), (2, 5), (3, 5), (4, 5)]
covers(x, y)

Returns True if y covers x and False otherwise.

EXAMPLES:

sage: Q = Poset([[1,5],[2,6],[3],[4],[],[6,3],[4]])
sage: Q.covers(Q(1),Q(6))
True
sage: Q.covers(Q(1),Q(4))
False
coxeter_transformation()

Returns 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.

EXAMPLES:

sage: M = Posets.PentagonPoset()._hasse_diagram.coxeter_transformation(); M
[ 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]

TESTS:

sage: M = Posets.PentagonPoset()._hasse_diagram.coxeter_transformation()
sage: M**8 == 1
True
dual()

Returns a poset that is dual to the given poset.

EXAMPLES:

sage: P = Posets.IntegerPartitions(4)
sage: H = P._hasse_diagram; H
Hasse diagram of a poset containing 5 elements
sage: H.dual()
Hasse diagram of a poset containing 5 elements

TESTS:

sage: H = Posets.IntegerPartitions(4)._hasse_diagram
sage: H.is_isomorphic( H.dual().dual() )
True
sage: H.is_isomorphic( H.dual() )
False
has_bottom()

Returns True if the poset has a unique minimal element.

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
has_top()

Returns True if the poset contains 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:[1],1:[]})
sage: Q.has_top()
True
interval(x, y)

Return a list of the elements \(z\) of self such that \(x \leq z \leq y\). The order is that induced by the ordering in self.linear_extension.

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: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram(dag)
sage: I = set([2,5,6,4,7])
sage: I == set(H.interval(2,7))
True
is_bounded()

Returns True if the poset contains a unique maximal element and a unique minimal element, and False otherwise.

EXAMPLES:

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

Returns True if the poset is totally ordered, and False otherwise.

EXAMPLES:

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

TESTS:

Check trac ticket #15330:

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

Returns True if self is the Hasse diagram of a complemented lattice, and False otherwise.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,2,3],1:[4],2:[4],3:[4]})
sage: H.is_complemented_lattice()
True

sage: H = HasseDiagram({0:[1,2],1:[3],2:[3],3:[4]})
sage: H.is_complemented_lattice()
False
is_distributive_lattice()

Returns True if self is the Hasse diagram of a distributive lattice, and False otherwise.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: H.is_distributive_lattice()
False
sage: H = HasseDiagram({0:[1,2],1:[3],2:[3]})
sage: H.is_distributive_lattice()
True
sage: H = HasseDiagram({0:[1,2,3],1:[4],2:[4],3:[4]})
sage: H.is_distributive_lattice()
False
is_gequal(x, y)

Returns True if x is greater than or equal to y, and False otherwise.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: Q = HasseDiagram({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = 0,1,4
sage: Q.is_gequal(x,y)
False
sage: Q.is_gequal(y,x)
False
sage: Q.is_gequal(x,z)
False
sage: Q.is_gequal(z,x)
True
sage: Q.is_gequal(z,y)
True
sage: Q.is_gequal(z,z)
True
is_graded()

Returns True if the poset is graded, and False otherwise.

A poset is graded if it admits a rank function. For more information about the rank function, see rank_function() and is_ranked().

EXAMPLES:

sage: P = Poset([[1],[2],[3],[4],[]])
sage: P.is_graded()
True
sage: Q = Poset([[1,5],[2,6],[3],[4],[],[6,3],[4]])
sage: Q.is_graded()
False
is_greater_than(x, y)

Returns True if x is greater than but not equal to y, and False otherwise.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: Q = HasseDiagram({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = 0,1,4
sage: Q.is_greater_than(x,y)
False
sage: Q.is_greater_than(y,x)
False
sage: Q.is_greater_than(x,z)
False
sage: Q.is_greater_than(z,x)
True
sage: Q.is_greater_than(z,y)
True
sage: Q.is_greater_than(z,z)
False
is_join_semilattice()

Returns True if self has a join operation, and False otherwise.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: H.is_join_semilattice()
True
sage: H = HasseDiagram({0:[2,3],1:[2,3]})
sage: H.is_join_semilattice()
False
sage: H = HasseDiagram({0:[2,3],1:[2,3],2:[4],3:[4]})
sage: H.is_join_semilattice()
False
is_lequal(i, j)

Returns True if i is less than or equal to j in the poset, and False otherwise.

Note

If the lequal_matrix() has been computed, then this method is redefined to use the cached matrix (see _alternate_is_lequal()).

TESTS:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = 0, 1, 4
sage: H.is_lequal(x,y)
False
sage: H.is_lequal(y,x)
False
sage: H.is_lequal(x,z)
True
sage: H.is_lequal(y,z)
True
sage: H.is_lequal(z,z)
True
is_less_than(x, y)

Returns True if x is less than or equal to y in the poset, and False otherwise.

TESTS:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[2], 1:[2], 2:[3], 3:[4], 4:[]})
sage: x,y,z = 0, 1, 4
sage: H.is_less_than(x,y)
False
sage: H.is_less_than(y,x)
False
sage: H.is_less_than(x,z)
True
sage: H.is_less_than(y,z)
True
sage: H.is_less_than(z,z)
False
is_linear_extension(lin_ext=None)

TESTS:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,2],1:[3],2:[3],3:[]})
sage: H.is_linear_extension(range(4))
True
sage: H.is_linear_extension([3,2,1,0])
False
is_meet_semilattice()

Returns True if self has a meet operation, and False otherwise.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: H.is_meet_semilattice()
True

sage: H = HasseDiagram({0:[1,2],1:[3],2:[3],3:[]})
sage: H.is_meet_semilattice()
True

sage: H = HasseDiagram({0:[2,3],1:[2,3]})
sage: H.is_meet_semilattice()
False
is_ranked()

Returns True if the poset is ranked, and False otherwise.

A poset is ranked if it admits a rank function. For more information about the rank function, see rank_function() and is_graded().

EXAMPLES:

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

Returns the matrix of joins of self. The (x,y)-entry of this matrix is the join of x and y in self.

This algorithm is modelled after the algorithm of Freese-Jezek-Nation (p217). It can also be found on page 140 of [Gec81].

Note

Once the matrix has been computed, it is stored in _join_matrix(). Delete this attribute if you want to recompute the matrix.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: H.join_matrix()
[0 1 2 3 4 5 6 7]
[1 1 4 7 4 7 7 7]
[2 4 2 6 4 5 6 7]
[3 7 6 3 7 7 6 7]
[4 4 4 7 4 7 7 7]
[5 7 5 7 7 5 7 7]
[6 7 6 6 7 7 6 7]
[7 7 7 7 7 7 7 7]

TESTS:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[2,3],1:[2,3]})
sage: H.join_matrix()
Traceback (most recent call last):
...
ValueError: Not a join-semilattice: no top element.

sage: H = HasseDiagram({0:[2,3],1:[2,3],2:[4],3:[4]})
sage: H.join_matrix()
Traceback (most recent call last):
...
ValueError: No join for x=...
lequal_matrix()

Returns the matrix whose (i,j) entry is 1 if i is less than j in the poset, and 0 otherwise; and redefines __lt__ to use this matrix.

EXAMPLES:

sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]])
sage: H = P._hasse_diagram
sage: H.lequal_matrix()
[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]

TESTS:

sage: H.lequal_matrix().is_immutable()
True
linear_extension()

TESTS:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,2],1:[3],2:[3],3:[]})
sage: H.linear_extension()
[0, 1, 2, 3]
linear_extensions()

TESTS:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,2],1:[3],2:[3],3:[]})
sage: H.linear_extensions()
[[0, 1, 2, 3], [0, 2, 1, 3]]
lower_covers_iterator(element)

Returns the list of elements that are covered by element.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: list(H.lower_covers_iterator(0))
[]
sage: list(H.lower_covers_iterator(4))
[1, 2]
maximal_elements()

Returns a 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]
meet_matrix()

Returns the matrix of meets of self. The (x,y)-entry of this matrix is the meet of x and y in self.

This algorithm is modelled after the algorithm of Freese-Jezek-Nation (p217). It can also be found on page 140 of [Gec81].

Note

Once the matrix has been computed, it is stored in _meet_matrix(). Delete this attribute if you want to recompute the matrix.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: H.meet_matrix()
[0 0 0 0 0 0 0 0]
[0 1 0 0 1 0 0 1]
[0 0 2 0 2 2 2 2]
[0 0 0 3 0 0 3 3]
[0 1 2 0 4 2 2 4]
[0 0 2 0 2 5 2 5]
[0 0 2 3 2 2 6 6]
[0 1 2 3 4 5 6 7]

REFERENCE:

[Gec81](1, 2) Fundamentals of Computation Theory Gecseg, F. Proceedings of the 1981 International Fct-Conference Szeged, Hungaria, August 24-28, vol 117 Springer-Verlag, 1981

TESTS:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[2,3],1:[2,3]})
sage: H.meet_matrix()
Traceback (most recent call last):
...
ValueError: Not a meet-semilattice: no bottom element.

sage: H = HasseDiagram({0:[1,2],1:[3,4],2:[3,4]})
sage: H.meet_matrix()
Traceback (most recent call last):
...
ValueError: No meet for x=...
minimal_elements()

Returns a 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
mobius_function(i, j)

Returns the value of the M”obius function of the poset on the elements i and j.

EXAMPLES:

sage: P = Poset([[1,2,3],[4],[4],[4],[]])
sage: H = P._hasse_diagram
sage: H.mobius_function(0,4)
2
sage: for u,v in P.cover_relations_iterator():
...    if P.mobius_function(u,v) != -1:
...        print "Bug in mobius_function!"
mobius_function_matrix()

Returns the matrix of the Mobius function of this poset

This returns the sparse matrix over \(\ZZ\) whose (x, y) entry is the value of the M”obius function of self evaluated on x and y, and redefines mobius_function() to use it.

Note

The result is cached in _mobius_function_matrix().

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: H.mobius_function_matrix()
[ 1 -1 -1 -1  1  0  1  0]
[ 0  1  0  0 -1  0  0  0]
[ 0  0  1  0 -1 -1 -1  2]
[ 0  0  0  1  0  0 -1  0]
[ 0  0  0  0  1  0  0 -1]
[ 0  0  0  0  0  1  0 -1]
[ 0  0  0  0  0  0  1 -1]
[ 0  0  0  0  0  0  0  1]

TESTS:

sage: H.mobius_function_matrix().is_immutable()
True
sage: hasattr(H,'_mobius_function_matrix')
True

sage: H.mobius_function == H._mobius_function_from_matrix
True
open_interval(x, y)

Return a list of the elements \(z\) of self such that \(x < z < y\). The order is that induced by the ordering in self.linear_extension.

EXAMPLES:

sage: uc = [[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]]
sage: dag = DiGraph(dict(zip(range(len(uc)),uc)))
sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram(dag)
sage: set([5,6,4]) == set(H.open_interval(2,7))
True
sage: H.open_interval(7,2)
[]
order_filter(elements)

Returns the order filter generated by a list of elements.

\(I\) is an order filter if, for any \(x\) in \(I\) and \(y\) such that \(y \ge x\), then \(y\) is in \(I\).

EXAMPLES:

sage: H = Posets.BooleanLattice(4)._hasse_diagram
sage: H.order_filter([3,8])
[3, 7, 8, 9, 10, 11, 12, 13, 14, 15]
order_ideal(elements)

Returns the order ideal generated by a list of elements.

\(I\) is an order ideal if, for any \(x\) in \(I\) and \(y\) such that \(y \le x\), then \(y\) is in \(I\).

EXAMPLES:

sage: H = Posets.BooleanLattice(4)._hasse_diagram
sage: H.order_ideal([7,10])
[0, 1, 2, 3, 4, 5, 6, 7, 8, 10]
plot(label_elements=True, element_labels=None, label_font_size=12, label_font_color='black', layout='acyclic', **kwds)

Returns a Graphics object corresponding to the Hasse diagram.

EXAMPLES:

sage: uc = [[2,3], [], [1], [1], [1], [3,4]]
sage: elm_lbls = Permutations(3).list()
sage: P = Poset(uc,elm_lbls)
sage: H = P._hasse_diagram
sage: levels = H.level_sets()
sage: heights = dict([[i, levels[i]] for i in range(len(levels))])
sage: type(H.plot(label_elements=True))
<class 'sage.plot.graphics.Graphics'>
sage: P = Posets.SymmetricGroupBruhatIntervalPoset([1,2,3,4], [3,4,1,2])
sage: P._hasse_diagram.plot()
principal_order_filter(i)

Returns the order filter generated by i.

EXAMPLES:

sage: H = Posets.BooleanLattice(4)._hasse_diagram
sage: H.principal_order_filter(2)
[2, 3, 6, 7, 10, 11, 14, 15]
principal_order_ideal(i)

Returns the order ideal generated by \(i\).

EXAMPLES:

sage: H = Posets.BooleanLattice(4)._hasse_diagram
sage: H.principal_order_ideal(6)
[0, 2, 4, 6]
rank(element=None)

Returns the rank of element, 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.)

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: H.rank(5)
2
sage: H.rank()
3
sage: Q = HasseDiagram({0:[1,2],1:[3],2:[],3:[]})
sage: Q.rank()
2
sage: Q.rank(1)
1
rank_function()

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([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]])
sage: P.rank_function() is not None
True
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

test for ticket trac ticket #14006:

sage: H = Poset()._hasse_diagram
sage: s = dumps(H)
sage: f = H.rank_function()
sage: s = dumps(H)
show(label_elements=True, element_labels=None, label_font_size=12, label_font_color='black', vertex_size=300, vertex_colors=None, **kwds)

Shows the Graphics object corresponding to the Hasse diagram. Optionally, it is labelled.

INPUT:

  • label_elements - whether to display element labels
  • element_labels - a dictionary of element labels

EXAMPLES:

sage: uc = [[2,3], [], [1], [1], [1], [3,4]]
sage: elm_lbls = Permutations(3).list()
sage: P = Poset(uc,elm_lbls)
sage: H = P._hasse_diagram
sage: levels = H.level_sets()
sage: heights = dict([[i, levels[i]] for i in range(len(levels))])
sage: H.show(label_elements=True)
top()

Returns the top 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
upper_covers_iterator(element)

Returns the list of elements that cover element.

EXAMPLES:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0:[1,3,2],1:[4],2:[4,5,6],3:[6],4:[7],5:[7],6:[7],7:[]})
sage: list(H.upper_covers_iterator(0))
[1, 2, 3]
sage: list(H.upper_covers_iterator(7))
[]

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