Finite posets¶
This module implements finite partially ordered sets. It defines:
FinitePoset 
A class for finite posets 
FinitePosets_n 
A class for finite posets up to isomorphism (i.e. unlabeled posets) 
Poset() 
Construct a finite poset from various forms of input data. 
is_poset() 
Return True if a directed graph is acyclic and transitively reduced. 
List of Poset methods¶
Comparing, intervals and relations
is_less_than() 
Return True if \(x\) is strictly less than \(y\) in the poset. 
is_greater_than() 
Return True if \(x\) is strictly greater than \(y\) in the poset. 
is_lequal() 
Return True if \(x\) is less than or equal to \(y\) in the poset. 
is_gequal() 
Return True if \(x\) is greater than or equal to \(y\) in the poset. 
compare_elements() 
Compare two element of the poset. 
closed_interval() 
Return the list of elements in a closed interval of the poset. 
open_interval() 
Return the list of elements in an open interval of the poset. 
relations() 
Return the list of relations in the poset. 
relations_iterator() 
Return an iterator over relations in the poset. 
order_filter() 
Return the upper set generated by elements. 
order_ideal() 
Return the lower set generated by elements. 
Covering
covers() 
Return True if y covers x . 
lower_covers() 
Return elements covered by given element. 
upper_covers() 
Return elements covering given element. 
cover_relations() 
Return the list of cover relations. 
lower_covers_iterator() 
Return an iterator over elements covered by given element. 
upper_covers_iterator() 
Return an iterator over elements covering given element. 
cover_relations_iterator() 
Return an iterator over cover relations of the poset. 
Properties of the poset
cardinality() 
Return the number of elements in the poset. 
height() 
Return the number of elements in a longest chain of the poset. 
width() 
Return the number of elements in a longest antichain of the poset. 
relations_number() 
Return the number of relations in the poset. 
dimension() 
Return the dimension of the poset. 
jump_number() 
Return the jump number of the poset. 
has_bottom() 
Return True if the poset has a unique minimal element. 
has_top() 
Return True if the poset has a unique maximal element. 
is_bounded() 
Return True if the poset has both unique minimal and unique maximal element. 
is_chain() 
Return True if the poset is totally ordered. 
is_connected() 
Return True if the poset is connected. 
is_graded() 
Return True if all maximal chains of the poset has same length. 
is_ranked() 
Return True if the poset has a rank function. 
is_rank_symmetric() 
Return True if the poset is rank symmetric. 
is_series_parallel() 
Return True if the poset can be built by ordinal sums and disjoint unions. 
is_greedy() 
Return True if all greedy linear extensions have equal number of jumps. 
is_jump_critical() 
Return True if removal of any element reduces the jump number. 
is_eulerian() 
Return True if the poset is Eulerian. 
is_incomparable_chain_free() 
Return True if the poset is (m+n)free. 
is_slender() 
Return True if the poset is slender. 
is_sperner() 
Return True if the poset is Sperner. 
is_join_semilattice() 
Return True is the poset has a join operation. 
is_meet_semilattice() 
Return True if the poset has a meet operation. 
Minimal and maximal elements
bottom() 
Return the bottom element of the poset, if it exists. 
top() 
Return the top element of the poset, if it exists. 
maximal_elements() 
Return the list of the maximal elements of the poset. 
minimal_elements() 
Return the list of the minimal elements of the poset. 
New posets from old ones
disjoint_union() 
Return the disjoint union of the poset with other poset. 
ordinal_sum() 
Return the ordinal sum of the poset with other poset. 
product() 
Return the Cartesian product of the poset with other poset. 
ordinal_product() 
Return the ordinal product of the poset with other poset. 
rees_product() 
Return the Rees product of the poset with other poset. 
lexicographic_sum() 
Return the lexicographic sum of posets. 
star_product() 
Return the star product of the poset with other poset. 
with_bounds() 
Return the poset with bottom and top element adjoined. 
without_bounds() 
Return the poset with bottom and top element removed. 
dual() 
Return the dual of the poset. 
completion_by_cuts() 
Return the DedekindMacNeille completion of the poset. 
intervals_poset() 
Return the poset of intervals of the poset. 
connected_components() 
Return the connected components of the poset as subposets. 
factor() 
Return the decomposition of the poset as a Cartesian product. 
ordinal_summands() 
Return the ordinal summands of the poset. 
subposet() 
Return the subposet containing elements with partial order induced by this poset. 
random_subposet() 
Return a random subposet that contains each element with given probability. 
relabel() 
Return a copy of this poset with its elements relabelled. 
canonical_label() 
Return copy of the poset canonically (re)labelled to integers. 
Chains & antichains
is_chain_of_poset() 
Return True if elements in the given list are comparable. 
is_antichain_of_poset() 
Return True if elements in the given list are incomparable. 
chains() 
Return the chains of the poset. 
antichains() 
Return the antichains of the poset. 
maximal_chains() 
Return the maximal chains of the poset. 
maximal_antichains() 
Return the maximal antichains of the poset. 
antichains_iterator() 
Return an iterator over the antichains of the poset. 
random_maximal_chain() 
Return a random maximal chain. 
random_maximal_antichain() 
Return a random maximal antichain. 
Drawing
show() 
Display the Hasse diagram of the poset. 
plot() 
Return a Graphic object corresponding the Hasse diagram of the poset. 
graphviz_string() 
Return a representation in the DOT language, ready to render in graphviz. 
Comparing posets
is_isomorphic() 
Return True if both posets are isomorphic. 
is_induced_subposet() 
Return True if given poset is an induced subposet of this poset. 
Polynomials
chain_polynomial() 
Return the chain polynomial of the poset. 
characteristic_polynomial() 
Return the characteristic polynomial of the poset. 
f_polynomial() 
Return the fpolynomial of the poset. 
flag_f_polynomial() 
Return the flag fpolynomial of the poset. 
h_polynomial() 
Return the hpolynomial of the poset. 
flag_h_polynomial() 
Return the flag hpolynomial of the poset. 
order_polynomial() 
Return the order polynomial of the poset. 
zeta_polynomial() 
Return the zeta polynomial of the poset. 
kazhdan_lusztig_polynomial() 
Return the KazhdanLusztig polynomial of the poset. 
coxeter_polynomial() 
Return the characteristic polynomial of the Coxeter transformation. 
degree_polynomial() 
Return the generating polynomial of degrees of vertices in the Hasse diagram. 
p_partition_enumerator() 
Return a \(P\)partition enumerator of the poset. 
Polytopes
chain_polytope() 
Return the chain polytope of the poset. 
order_polytope() 
Return the order polytope of the poset. 
Graphs
hasse_diagram() 
Return the Hasse diagram of the poset as a directed graph. 
cover_relations_graph() 
Return the (undirected) graph of cover relations. 
comparability_graph() 
Return the comparability graph of the poset. 
incomparability_graph() 
Return the incomparability graph of the poset. 
frank_network() 
Return Frank’s network of the poset. 
linear_extensions_graph() 
Return the linear extensions graph of the poset. 
Linear extensions
is_linear_extension() 
Return True if the given list is a linear extension of the poset. 
linear_extension() 
Return a linear extension of the poset. 
linear_extensions() 
Return the enumerated set of all the linear extensions of the poset. 
promotion() 
Return the (extended) promotion on the linear extension of the poset. 
evacuation() 
Return evacuation on the linear extension associated to the poset. 
with_linear_extension() 
Return a copy of self with a different default linear extension. 
random_linear_extension() 
Return a random linear extension. 
Matrices
lequal_matrix() 
Computes the matrix whose (i,j) entry is 1 if self.linear_extension()[i] < self.linear_extension()[j] and 0 otherwise. 
moebius_function() 
Return the value of Möbius function of given elements in the poset. 
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] . 
coxeter_transformation() 
Return the matrix of the AuslanderReiten translation acting on the Grothendieck group of the derived category of modules. 
coxeter_smith_form() 
Return the Smith form of the Coxeter transformation. 
Miscellanous
sorted() 
Return given list sorted by the poset. 
isomorphic_subposets() 
Return all subposets isomorphic to another poset. 
isomorphic_subposets_iterator() 
Return an iterator over the subposets isomorphic to another poset. 
has_isomorphic_subposet() 
Return True if the poset contains a subposet isomorphic to another poset. 
list() 
List the elements of the poset. 
cuts() 
Return the cuts of the given poset. 
dilworth_decomposition() 
Return a partition of the points into the minimal number of chains. 
greene_shape() 
Computes the GreeneKleitman partition aka Greene shape of the poset self . 
incidence_algebra() 
Return the incidence algebra of self . 
is_EL_labelling() 
Return whether f is an EL labelling of the poset. 
isomorphic_subposets_iterator() 
Return an iterator over the subposets isomorphic to another poset. 
isomorphic_subposets() 
Return all subposets isomorphic to another poset. 
level_sets() 
Return elements grouped by maximal number of cover relations from a minimal element. 
order_complex() 
Return the order complex associated to this poset. 
random_order_ideal() 
Return a random order ideal of self with uniform probability. 
rank() 
Return the rank of an element, or the rank of the poset. 
rank_function() 
Return a rank function of the poset, if it exists. 
unwrap() 
Unwraps an element of this poset. 
Classes and functions¶

class
sage.combinat.posets.posets.
FinitePoset
(hasse_diagram, elements, category, facade, key)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
A (finite) \(n\)element poset constructed from a directed acyclic graph.
INPUT:
hasse_diagram
– an instance ofFinitePoset
, or aDiGraph
that is transitivelyreduced, acyclic, loopfree, and multiedgefree.elements
– an optional list of elements, withelement[i]
corresponding to vertexi
. Ifelements
isNone
, then it is set to be the vertex set of the digraph. Note that if this option is set, thenelements
is considered as a specified linear extension of the poset and the \(linear_extension\) attribute is set.category
–FinitePosets
, or a subcategory thereof.facade
– a boolean orNone
(default); whether theFinitePoset
’s elements should be wrapped to make them aware of the Poset they belong to. If
facade = True
, theFinitePoset
’s elements are exactly those given as input.  If
facade = False
, theFinitePoset
’s elements will becomePosetElement
objects.  If
facade = None
(default) the expected behaviour is the behaviour offacade = True
, unless the opposite can be deduced from the context (i.e. for instance if aFinitePoset
is built from anotherFinitePoset
, itself built withfacade = False
)
 If
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
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
isPosetElement
. It can also define_dual_class
which is the class of dual posets of this class. E.g.FiniteMeetSemilattice._dual_class
isFiniteJoinSemilattice
.
Element
¶

antichains
(element_constructor=<type 'list'>)¶ 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
PairwiseCompatibleSubsets
) of all antichains of the poset, each of which is given as anelement_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
sage.combinat.subsets_pairwise.PairwiseCompatibleSubsets
andSearchForest
. At this point, iterating through this set is about twice slower than usingantichains_iterator()
(tested onposets.AntichainPoset(15)
). The algorithm is the same (depth first search through the tree), butantichains_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.
See also

antichains_iterator
()¶ 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])
See also

bottom
()¶ 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
See also

canonical_label
(algorithm=None)¶ Return the unique poset on the labels \(\{0, \ldots, n1\}\) (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 meetsemilattice containing 3 elements
See also
 Canonical labeling of directed graphs:
canonical_label()

cardinality
()¶ Return the number of elements in the poset.
EXAMPLES:
sage: Poset([[1,2,3],[4],[4],[4],[]]).cardinality() 5
See also
degree_polynomial()
for a more refined invariant

chain_polynomial
()¶ 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 fvector 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 (
order_polynomial()
).See also
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

chain_polytope
()¶ 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
\[x(p) \geq 0 \mbox{ for all } p \in P,\]and
\[x(p_1) + x(p_2) + \ldots + x(p_k) \leq 1 \mbox{ for all chains } p_1 < p_2 < \ldots < p_k \mbox{ in } P.\]This polytope was defined and studied in [St1986].
EXAMPLES:
sage: P = posets.AntichainPoset(3) sage: Q = P.chain_polytope();Q A 3dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices sage: P = posets.PentagonPoset() sage: Q = P.chain_polytope();Q A 5dimensional polyhedron in ZZ^5 defined as the convex hull of 8 vertices

chains
(element_constructor=<type 'list'>, exclude=None)¶ Return the chains of the poset.
A chain of a poset is a set of elements of the poset that are pairwise comparable.
INPUT:
element_constructor
– a function taking an iterable as argument (default:list
)exclude
– elements of the poset to be excluded (default:None
)
OUTPUT:
The enumerated set (of type
PairwiseCompatibleSubsets
) of all chains of the poset, each of which is given as anelement_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
See also

characteristic_polynomial
()¶ 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
\[\sum_{x \in P} \mu(\hat{0}, x) q^{n\rho(x)} \in \ZZ[q],\]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

closed_interval
(x, y)¶ 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]
See also

comparability_graph
()¶ 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 article 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

compare_elements
(x, y)¶ 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
 If \(x < y\), return

completion_by_cuts
()¶ Return the completion by cuts of
self
.This is the smallest lattice containing the poset. This is also called the DedekindMacNeille completion.
See the Wikipedia article DedekindMacNeille 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
See also

connected_components
()¶ 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]]
See also

cover_relations
()¶ Return the list of pairs
[x, y]
of elements of the poset such thaty
coversx
.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]]

cover_relations_graph
()¶ 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)
See also

cover_relations_iterator
()¶ 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]]

covers
(x, y)¶ Return
True
ify
coversx
andFalse
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

coxeter_polynomial
()¶ 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
See also

coxeter_smith_form
(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
See also
coxeter_transformation()
,coxeter_matrix()

coxeter_transformation
()¶ Return the Coxeter transformation of the poset.
OUTPUT:
a square matrix with integer coefficients
The output is the matrix of the AuslanderReiten 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]
See also

cuts
()¶ 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: [list(c) for c in Pc] [[0], [], [0, 1, 2], [2], [1]] sage: Pc[0] frozenset({0})
See also

degree_polynomial
()¶ Return the generating polynomial of degrees of vertices in
self
.This is the sum
\[\sum_{v \in P} x^{\operatorname{in}(v)} y^{\operatorname{out}(v)},\]where
in(v)
andout(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
See also
cardinality()
for the value at \((x, y) = (1, 1)\)

dilworth_decomposition
()¶ 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 article Dilworth’s_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]]
See also
level_sets()
to return elements grouped to antichains.

dimension
(certificate=False)¶ Return the dimension of the Poset.
The (DushnikMiller) 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:
\[u\leq_P v\ \text{if and only if }\ \forall i, u\leq_{P_i} v\]For more information, see the Wikipedia article 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.
Note
The speed of this function greatly improves when more efficient MILP solvers (e.g. Gurobi, CPLEX) are installed. See
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 ticket #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
Mixed Integer Linear Program
.The problem is known to be NPcomplete.
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  architecturedependent [[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

disjoint_union
(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 elementv
in this poset will be named(0,v)
and each elementu
inother
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']]
See also

dual
()¶ 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 meetsemilattice is joinsemilattice 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 joinsemilattice containing 3 elements sage: A(2) < A(1) True
See also

evacuation
()¶ 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 \((\tau_1 \cdots \tau_{n1}) (\tau_1 \cdots \tau_{n2}) \cdots (\tau_1)\), to the default linear extension \(\pi\) ofself
(seeevacuation()
), and relabelingself
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]]
See also
linear_extension()
with_linear_extension()
and thelinear_extension
option ofPoset()
evacuation()
promotion()
AUTHOR:
 Anne Schilling (20120218)

f_polynomial
()¶ 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

factor
()¶ 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]
See also
REFERENCES:
[Feig1986] Joan Feigenbaum, Directed CartesianProduct Graphs have Unique Factorizations that can be computed in Polynomial Time, Discrete Applied Mathematics 15 (1986) 105110 doi:10.1016/0166218X(86)900235

flag_f_polynomial
()¶ 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
\[\begin{split}\sum_{\substack{p_0 < p_1 < \ldots < p_k, \\ p_0 = \min P, \ p_k = \max P}} x_{\rho(p_1)} x_{\rho(p_2)} \cdots x_{\rho(p_k)} \in \ZZ[x_1, x_2, \cdots, x_n]\end{split}\]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 \(x_0\) doesn’t 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 article hvector.
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
See also

flag_h_polynomial
()¶ 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
\[\prod_{k=1}^n (1x_k) \cdot f \left(\frac{x_1}{1x_1}, \frac{x_2}{1x_2}, \cdots, \frac{x_n}{1x_n}\right) \in \ZZ[x_1, x_2, \cdots, x_n],\]where \(f\) is the flag \(f\)polynomial of \(P\) (see
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 article hvector.
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
See also

frank_network
()¶ 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
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 \(2P + 2\) 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 (20130509)

ge
(x, y)¶ Return
True
if \(x\) is greater than or equal to \(y\) in the poset, andFalse
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
See also

graphviz_string
(graph_string='graph', edge_string='')¶ Returns 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"; }

greene_shape
()¶ Return the GreeneKleitman partition of
self
.The GreeneKleitman 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 (20130509)

gt
(x, y)¶ Return
True
if \(x\) is greater than but not equal to \(y\) in the poset, andFalse
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 nonfacade posets also
>
works:sage: P = Poset({3: [1, 2]}, facade=False) sage: P(2) > P(3) True
See also

h_polynomial
()¶ 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:
\[h(q) = (1q)^{\deg f} f \left( \frac{q}{1q} \right),\]where \(f\) and \(h\) denote the \(f\)polynomial and the \(h\)polynomial, respectively.
See Wikipedia article hvector.
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

has_bottom
()¶ Return
True
if the poset has a unique minimal element, andFalse
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
See also
 Dual Property:
has_top()
 Stronger properties:
is_bounded()
 Other:
bottom()
 Dual Property:

has_isomorphic_subposet
(other)¶ Return
True
if the poset contains a subposet isomorphic toother
.By subposet we mean that there exist a set
X
of elements such thatself.subposet(X)
is isomorphic toother
.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

has_top
()¶ Return
True
if the poset has a unique maximal element, andFalse
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
See also
 Dual Property:
has_bottom()
 Stronger properties:
is_bounded()
 Other:
top()
 Dual Property:

hasse_diagram
()¶ Return the Hasse diagram of the poset as a Sage
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 thedot2tex
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)]

height
(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)
, whereh
is the height andc
is a chain of maximum cardinality. Ifcertificate=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])

incidence_algebra
(R, prefix='I')¶ Return the incidence algebra of
self
overR
.OUTPUT:
An instance of
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

incomparability_graph
()¶ 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
See also

interval
(x, y)¶ Return a list of the elements \(z\) such that \(x \le z \le y\).
INPUT:
x
– any element of the posety
– 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]

intervals_number
()¶ 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
See also

intervals_poset
()¶ 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 posetmorphisms from the 2chain 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

is_EL_labelling
(f, return_raising_chains=False)¶ Return
True
iff
is an EL labelling ofself
.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 elementsa
andb
inself
such thatb
coversa
and returning elements in a totally ordered set.return_raising_chains
(optional; default:False
) ifTrue
, returns the set of all raising chains inself
, 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]}

is_antichain_of_poset
(elms)¶ Return
True
ifelms
is an antichain of the poset andFalse
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

is_bounded
()¶ Return
True
if the poset is bounded, andFalse
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
See also
 Weaker properties:
has_bottom()
,has_top()
 Other:
with_bounds()
,without_bounds()
 Weaker properties:

is_chain
()¶ Return
True
if the poset is totally ordered (“chain”), andFalse
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

is_chain_of_poset
(elms, ordered=False)¶ Return
True
ifelms
is a chain of the poset, andFalse
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 posetordered
– a Boolean. IfTrue
, then returnTrue
only if elements inelms
are strictly increasing in the poset; this makes no sense ifelms
is a set. IfFalse
(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

is_connected
()¶ Return
True
if the poset is connected, andFalse
otherwise.A poset is connected if it’s 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
See also

is_eulerian
(k=None, certificate=False)¶ Return
True
if the poset is Eulerian, andFalse
otherwise.The poset is expected to be graded and bounded.
A poset is Eulerian if every nontrivial interval has the same number of elements of even rank as of odd rank. A poset is \(k\)eulerian if every nontrivial interval up to rank \(k\) is Eulerian.
See Wikipedia article Eulerian_poset.
INPUT:
k
, an integer – only check if the poset is \(k\)eulerian. IfNone
(the default), check if the poset is Eulerian.certificate
, a Boolean – (default:False
) whether to return a certificate
OUTPUT:
 If
certificate=True
return eitherTrue, None
orFalse, (a, b)
, where the interval(a, b)
is not Eulerian. Ifcertificate=False
returnTrue
orFalse
.
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 4eulerian:
sage: P = Poset(DiGraph('[email protected][email protected][email protected][email protected][email protected][email protected]??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))

is_gequal
(x, y)¶ Return
True
if \(x\) is greater than or equal to \(y\) in the poset, andFalse
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
See also

is_graded
()¶ Return
True
if the poset is graded, andFalse
otherwise.A poset is graded if all its maximal chains have the same length.
There are various competing definitions for graded posets (see Wikipedia article 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
See also

is_greater_than
(x, y)¶ Return
True
if \(x\) is greater than but not equal to \(y\) in the poset, andFalse
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 nonfacade posets also
>
works:sage: P = Poset({3: [1, 2]}, facade=False) sage: P(2) > P(3) True
See also

is_greedy
(certificate=False)¶ Return
True
if the poset is greedy, andFalse
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. Ifcertificate=False
returnTrue
orFalse
.
EXAMPLES:
This is not a selfdual 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]))

is_incomparable_chain_free
(m, n=None)¶ Return
True
if the poset is \((m+n)\)free, andFalse
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]])
AUTHOR:
 Eric Rowland (20130528)

is_induced_subposet
(other)¶ Return
True
if the poset is an induced subposet ofother
, andFalse
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 ifother
directly contains the poset as an induced subposet. For isomorphic subposets seehas_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

is_isomorphic
(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

is_join_semilattice
(certificate=False)¶ Return
True
if the poset has a join operation, andFalse
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. Ifcertificate=False
returnTrue
orFalse
.
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))
See also
 Dual property:
is_meet_semilattice()
 Stronger properties:
is_lattice()

is_jump_critical
(certificate=False)¶ Return
True
if the poset is jumpcritical, andFalse
otherwise.A poset \(P\) is jumpcritical 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. Ifcertificate=False
returnTrue
orFalse
.
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)
See also

is_lequal
(x, y)¶ Return
True
if \(x\) is less than or equal to \(y\) in the poset, andFalse
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
See also

is_less_than
(x, y)¶ Return
True
if \(x\) is less than but not equal to \(y\) in the poset, andFalse
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 nonfacade posets also
<
works:sage: P = Poset({3: [1, 2]}, facade=False) sage: P(1) < P(2) False
See also

is_linear_extension
(l)¶ Returns whether
l
is a linear extension ofself
INPUT:
l
– a list (or iterable) containing all of the elements ofself
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
LinearExtensionsOfPosets
See also

is_meet_semilattice
(certificate=False)¶ Return
True
if the poset has a meet operation, andFalse
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. Ifcertificate=False
returnTrue
orFalse
.
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)))
See also
 Dual property:
is_join_semilattice()
 Stronger properties:
is_lattice()

is_parent_of
(x)¶ Return
True
ifx
is an element of the poset.

is_rank_symmetric
()¶ Return
True
if the poset is rank symmetric, andFalse
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 \(hi\) 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

is_ranked
()¶ Return
True
if the poset is ranked, andFalse
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
See also

is_series_parallel
()¶ Return
True
if the poset is seriesparallel, andFalse
otherwise.A poset is seriesparallel if it can be built up from oneelement posets using the operations of disjoint union and ordinal sum. This is also called Nfree property: every poset that is not seriesparallel contains a subposet isomorphic to the 4element Nshaped poset where \(a > c, d\) and \(b > d\).
Note
Some papers use the term Nfree for posets having no Nshaped poset as a coverpreserving subposet. This definition is not used here.
See Wikipedia article Seriesparallel 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

is_slender
(certificate=False)¶ Return
True
if the poset is slender, andFalse
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. Ifcertificate=False
returnTrue
orFalse
.
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)))

is_sperner
()¶ Return
True
if the poset is Sperner, andFalse
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 article Sperner_property_of_a_partially_ordered_set
See also
EXAMPLES:
sage: posets.SetPartitions(3).is_sperner() True sage: P = Poset({0:[3,4,5],1:[5],2:[5]}) sage: P.is_sperner() False

isomorphic_subposets
(other)¶ Return a list of subposets of
self
isomorphic toother
.By subposet we mean
self.subposet(X)
which is isomorphic toother
and whereX
is a subset of elements ofself
.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
isomorphic_subposets_iterator()
.

isomorphic_subposets_iterator
(other)¶ Return an iterator over the subposets of
self
isomorphic toother
.By subposet we mean
self.subposet(X)
which is isomorphic toother
and whereX
is a subset of elements ofself
.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
subgraph_search_iterator()
returning labelled subgraphs. On the other hand, this function does not eat memory likeisomorphic_subposets()
does.

jump_number
(certificate=False)¶ 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. Ifcertificate=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 NPcomplete.
See also

kazhdan_lusztig_polynomial
(x=None, y=None, q=None, canonical_labels=None)¶ Return the KazhdanLusztig 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 KazhdanLusztig polynomial of \(G\) as the unique polynomial \(P_G(q)\) satisfying the following:
If \(\operatorname{rank} G = 0\), then \(P_G(q) = 1\).
If \(\operatorname{rank} G > 0\), then \(\deg P_G(q) < \frac{1}{2} \operatorname{rank} G\).
We have
\[q^{\operatorname{rank} G} P_G(q^{1}) = \sum_{x \in G} \chi_{I_x}(q) P_{F^x}(q).\]
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 isTrue
ifx
andy
are both not specified andFalse
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 (27122014)

le
(x, y)¶ Return
True
if \(x\) is less than or equal to \(y\) in the poset, andFalse
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
See also

lequal_matrix
(ring=Integer Ring, sparse=False)¶ Compute the matrix whose
(i,j)
entry is 1 ifself.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

level_sets
()¶ Return elements grouped by maximal number of cover relations from a minimal element.
This returns a list of lists
l
such thatl[i]
is the set of minimal elements of the poset obtained by removing the elements inl[0], l[1], ..., l[i1]
. (In particular,l[0]
is the set of minimal elements ofself
.)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]]
See also
dilworth_decomposition()
to return elements grouped to chains.

lexicographic_sum
(P)¶ 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.
disjoint_union()
andordinal_sum()
are special cases of lexicographic sum where the index poset is an (anti)chain.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)]

linear_extension
(linear_extension=None, check=True)¶ Return a linear extension of this poset.
A linear extension of a finite poset \(P\) of size \(n\) is a total ordering \(\pi := \pi_0 \pi_1 \ldots \pi_{n1}\) 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 ofself
check
– a boolean (default: True); whether to check thatlinear_extension
is indeed a linear extension ofself
.
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 fullfeatured 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]
See also
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

linear_extensions
(facade=False)¶ Returns 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 listsWarning
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 ofL
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'>
See also

linear_extensions_graph
()¶ 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

list
()¶ List the elements of the poset. This just returns the result of
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'>

lower_covers
(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) []
See also

lower_covers_iterator
(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

lt
(x, y)¶ Return
True
if \(x\) is less than but not equal to \(y\) in the poset, andFalse
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 nonfacade posets also
<
works:sage: P = Poset({3: [1, 2]}, facade=False) sage: P(1) < P(2) False
See also

maximal_antichains
()¶ 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]]
See also

maximal_chains
(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
Returns list of the maximal chains of this poset.
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]]
See also

maximal_elements
()¶ 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]
See also

minimal_elements
()¶ 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
See also

moebius_function
(x, y)¶ 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

moebius_function_matrix
(ring=Integer Ring, sparse=False)¶ Return a matrix whose
(i,j)
entry is the value of the Möbius function evaluated atself.linear_extension()[i]
andself.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

open_interval
(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]
See also

order_complex
(on_ints=False)¶ 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
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)}

order_filter
(elements)¶ 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]
See also

order_ideal
(elements)¶ 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]
See also

order_ideal_plot
(elements)¶ 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

order_polynomial
()¶ 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 orderpreserving 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]
See also

order_polytope
()¶ 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
\[0 \leq x(p) \leq 1 \mbox{ for all } p \in P,\]and
\[x(p) \leq x(q) \mbox{ for all } p, q \in P \mbox{ satisfying } p < q.\]This polytope was defined and studied in [St1986].
EXAMPLES:
sage: P = posets.AntichainPoset(3) sage: Q = P.order_polytope();Q A 3dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices sage: P = posets.PentagonPoset() sage: Q = P.order_polytope();Q A 5dimensional 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

ordinal_product
(other, labels='pairs')¶ Return the ordinal product of
self
andother
.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 posetlabels
– 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')]]
See also

ordinal_sum
(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 joinsemilattices.
INPUT:
other
, a poset.labels
 (defaults to ‘pairs’) If set to ‘pairs’, each elementv
in this poset will be named(0,v)
and each elementu
inother
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 joinsemilattice 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 joinsemilattice containing 4 elements sage: L.ordinal_sum(L) Finite lattice containing 4 elements

ordinal_summands
()¶ Return the ordinal summands of the poset as subposets.
The ordinal summands of a poset \(P\) is the longest list of nonempty 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
See also
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.

p_partition_enumerator
(tup, R, weights=None, check=False)¶ 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 ofP
. Then, the generalized \(P\)partition enumerator corresponding to weightsweights
is \(\sum_f \prod_{p \in P} x_{f(p)}^{w(p)}\), where the sum is again over all \(P\)partitions \(f\). Here, \(w(p)\) isweights[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 ringweights
– (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 totup
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]

plot
(label_elements=True, element_labels=None, layout='acyclic', cover_labels=None, **kwds)¶ 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 elementselement_color
 a color for elements not set inelement_colors
element_shape
 the shape of elements, like's'
for square; see https://matplotlib.org/api/markers_api.html for the listelement_size
(default: 200)  the size of elementslabel_elements
(default:True
)  whether to display element labelselement_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 elementscover_color
 a color for elements not set incover_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 toNone
(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 plottitle
 a title for the plotfontsize
 fontsize for the titleborder
(default:False
)  whether to draw a border over the plot
Note
All options of
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
 Options to change element look:

product
(other)¶ 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
See also

promotion
(i=1)¶ 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 \(\tau_i \tau_{i+1} \cdots \tau_{n1}\) to the default linear extension \(\pi\) ofself
(seepromotion()
), and relabelingself
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 \(1\) fromself
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]]
See also
linear_extension()
with_linear_extension()
and thelinear_extension
option ofPoset()
promotion()
evacuation()
AUTHOR:
 Anne Schilling (20120218)

random_linear_extension
()¶ 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]

random_maximal_antichain
()¶ 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]

random_maximal_chain
()¶ 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]

random_order_ideal
(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 ifdirection='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() [0, 1, 2, 3, 4, 5, 6] sage: P.random_order_ideal(direction='up') [6, 7] sage: P.random_order_ideal(direction='antichain') [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

random_subposet
(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]]

rank
(element=None)¶ Return the rank of an element
element
in the posetself
, or the rank of the poset ifelement
isNone
.(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)]

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(([1,2,3,4],[[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

rees_product
(other)¶ Return the Rees product of
self
andother
.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
See also

relabel
(relabeling=None)¶ 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 (nonwrapped) 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 \(n1\) 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 joinsemilattice containing 2 elements
Note
As can be seen in the above examples, the default linear extension of
Q
is that ofP
after relabeling. In particular,P
andQ
share the same internal Hasse diagram.

relations
()¶ 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]]
See also
AUTHOR:
 Rob Beezer (20110504)

relations_iterator
(strict=False)¶ 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 (defaultFalse
) ifTrue
, 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]]
See also
AUTHOR:
 Rob Beezer (20110504)

relations_number
()¶ 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
See also

show
(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 labelselement_labels
(default:None
)  a dictionary of element labelscover_labels
 a dictionary, list or function representing labels of the covers ofself
. When set toNone
(default) no label is displayed on the edges of the Hasse Diagram.
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)

sorted
(l, allow_incomparable=True, remove_duplicates=False)¶ Return the list \(l\) sorted by the poset.
INPUT:
l
– a list of elements of the posetallow_incomparable
– a Boolean. IfTrue
(the default), return incomparable elements in some order; ifFalse
, raise an error ifl
is not a chain of the poset.remove_duplicates
 a Boolean. IfTrue
, 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

star_product
(other, labels='pairs')¶ 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 elementv
in this poset will be named(0, v)
and each elementu
inother
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]

subposet
(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 nonfacade poset is again a nonfacade poset:
sage: P = posets.PentagonPoset(facade=False) sage: Q = P.subposet([0, 1, 2, 4]) sage: Q(1) < Q(2) False

top
()¶ 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

unwrap
(element)¶ Return the element
element
of the posetself
in unwrapped form.INPUT:
element
– an element ofself
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 don’t know if
P
is a facade poset or not.

upper_covers
(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]
See also

upper_covers_iterator
(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'>

width
(certificate=False)¶ Return the width of the poset (the size of its longest antichain).
It is computed through a matching in a bipartite graph; see Wikipedia article Dilworth’s_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. Ifcertificate=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]

with_bounds
(labels=('bottom', 'top'))¶ 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 beNone
, 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
See also
without_bounds()
for the reverse operation

with_linear_extension
(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]]
Note
With the current implementation, this requires relabeling the internal
DiGraph
which is \(O(n+m)\), where \(n\) is the number of elements and \(m\) the number of cover relations.

without_bounds
()¶ Return the poset without its top and bottom elements.
This is useful as an input for the method
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() []
See also
with_bounds()
for the reverse operation

zeta_polynomial
()¶ 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 \(x_1 \leq x_2 \leq \dots \leq x_{m1}\) of elements of the poset.
The polynomial \(Z(q)\) is integralvalued, but generally does not have integer coefficients. It can be computed as
\[Z(q) = \sum_{k \geq 1} \dbinom{q2}{k1} c_k,\]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

class
sage.combinat.posets.posets.
FinitePosets_n
(n)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
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]]

cardinality
(from_iterator=False)¶ Return the cardinality of this object.
Note
By default, this returns precomputed values obtained from the OnLine Encyclopedia of Integer Sequences (OEIS sequence 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


sage.combinat.posets.posets.
Poset
(data=None, element_labels=None, cover_relations=False, linear_extension=False, category=None, facade=None, key=None)¶ Construct a finite poset from various forms of input data.
INPUT:
data
– different input are accepted by this constructor:A twoelement list or tuple
(E, R)
, whereE
is a collection of elements of the poset andR
is a collection of relationsx <= y
, each represented as a twoelement list/tuple/iterable such as[x, y]
. The poset is then the transitive closure of the provided relations. Ifcover_relations=True
, thenR
is assumed to contain exactly the cover relations of the poset. IfE
is empty, thenE
is taken to be the set of elements appearing in the relationsR
.A twoelement list or tuple
(E, f)
, whereE
is the set of elements of the poset andf
is a function such that, for any pairx, y
of elements ofE
,f(x, y)
returns whetherx <= y
. Ifcover_relations=True
, thenf(x, y)
should instead return whetherx
is covered byy
.A dictionary of upper covers:
data[x]
is a list of the elements that cover the element \(x\) in the poset.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.
An acyclic, loopfree and multiedge free
DiGraph
. Ifcover_relations
isTrue
, then the edges of the digraph are assumed to correspond to the cover relations of the poset. Otherwise, the cover relations are computed.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)
, thenE
is taken to be the linear extension.facade
– a boolean orNone
(default); whether thePoset()
’s elements should be wrapped to make them aware of the Poset they belong to. If
facade = True
, thePoset()
’s elements are exactly those given as input.  If
facade = False
, thePoset()
’s elements will becomePosetElement
objects.  If
facade = None
(default) the expected behaviour is the behaviour offacade = True
, unless the opposite can be deduced from the context (i.e. for instance if aPoset()
is built from anotherPoset()
, itself built withfacade = False
)
 If
OUTPUT:
FinitePoset
– an instance of theFinitePoset
class.If
category
is specified, then the poset is created in this category instead ofFinitePosets
.See also
EXAMPLES:
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 noncover 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]]
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
A dictionary of upper covers:
sage: Poset({'a':['b','c'], 'b':['d'], 'c':['d'], 'd':[]}) Finite poset containing 4 elements
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 2element poset.
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
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.See also
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
DiGraph
is used to construct the poset, and the vertices of aDiGraph
are converted to plain Pythonint
’s if they areInteger
’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
Integer
’s if they areint
’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
Unique representation
As most parents,
Poset
have unique representation (seeUniqueRepresentation
). 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 toUniqueRepresentation
. It is otherwise ignored by the poset constructor.

sage.combinat.posets.posets.
is_poset
(dig)¶ Return
True
if a directed graph is acyclic and transitively reduced, andFalse
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