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_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_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. |
star_product() |
Return the star product of the poset with other poset. |
with_bounds() |
Return the poset with bottom and top element adjoined. |
dual() |
Return the dual of the poset. |
completion_by_cuts() |
Return the Dedekind-MacNeille 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. |
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 the given list is a chain of the poset. |
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. |
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 f-polynomial of the poset. |
flag_f_polynomial() |
Return the flag f-polynomial of the poset. |
h_polynomial() |
Return the h-polynomial of the poset. |
flag_h_polynomial() |
Return the flag h-polynomial 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 Kazhdan-Lusztig polynomial of the poset. |
coxeter_polynomial() |
Return the characteristic polynomial of the Coxeter transformation. |
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. |
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. |
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 Auslander-Reiten translation acting on the Grothendieck group of the derived category of modules. |
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 Greene-Kleitman partition aka Greene shape of the poset self . |
incidence_algebra() |
Return the indicence 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. |
lequal_matrix() |
Computes the matrix whose (i,j) entry is 1 if self.linear_extension()[i] < self.linear_extension()[j] and 0 otherwise. |
level_sets() |
Return a list l such that l[i+1] is the set of minimal elements of the poset obtained by removing the elements in l[0], l[1], ..., l[i]. |
order_complex() |
Return the order complex associated to this poset. |
p_partition_enumerator() |
Return a \(P\)-partition enumerator of the 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. |
with_linear_extension() |
Return a copy of self with a different default linear extension. |
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 transitively-reduced, acyclic, loop-free, and multiedge-free.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() Join of Category of finite posets and Category of finite enumerated sets 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() Join of Category of finite posets and Category of finite enumerated sets sage: parent(P[0]) is P True sage: Q = Poset(DiGraph({'a':['b'],'b':['c'],'c':['d']}), facade=True) sage: Q.category() Join of Category of finite posets and Category of finite enumerated sets and Category of facade sets 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() Join of Category of finite posets and Category of finite enumerated sets and Category of facade sets 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() Join of Category of finite posets and Category of finite enumerated sets 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
.TESTS:
Equality is derived from
UniqueRepresentation
. We check that this gives consistent results:sage: P = Poset([[1,2],[3],[3]]) sage: P == P True sage: Q = Poset([[1,2],[],[1]]) sage: Q == P False sage: p1, p2 = Posets(2).list() sage: p2 == p1, p1 != p2 (False, True) sage: [[p1.__eq__(p2) for p1 in Posets(2)] for p2 in Posets(2)] [[True, False], [False, True]] sage: [[p2.__eq__(p1) for p1 in Posets(2)] for p2 in Posets(2)] [[True, False], [False, True]] sage: [[p2 == p1 for p1 in Posets(3)] for p2 in Posets(3)] [[True, False, False, False, False], [False, True, False, False, False], [False, False, True, False, False], [False, False, False, True, False], [False, False, False, False, True]] sage: [[p1.__ne__(p2) for p1 in Posets(2)] for p2 in Posets(2)] [[False, True], [True, False]] sage: P = Poset([[1,2,4],[3],[3]]) sage: Q = Poset([[1,2],[],[1],[4]]) sage: P != Q True sage: P != P False sage: Q != Q False sage: [[p1.__ne__(p2) for p1 in Posets(2)] for p2 in Posets(2)] [[False, True], [True, False]] sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True) sage: Q = Poset(P) sage: Q == P False sage: Q = Poset(P, linear_extension=True) sage: Q == P True
-
Element
¶ alias of
PosetElement
-
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 \(\{0, \ldots, n-1\}\) (where \(n\) is the number of elements in the poset) that is isomorphic to this poset and invariant in the isomorphism class.
INPUT:
algorithm
, a string orNone
– a parameter forwarded to underlying graph function to select the algorithm to use
See also
- Canonical labeling of directed graphs:
canonical_label()
EXAMPLES:
sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True) sage: P.list() [1, 2, 3, 4, 6, 12] sage: Q = P.canonical_label() sage: sorted(Q.list()) [0, 1, 2, 3, 4, 5] sage: Q.is_isomorphic(P) True
Canonical labeling of (semi)lattice returns (semi)lattice:
sage: D = DiGraph({'a':['b','c']}) sage: P = Poset(D) sage: ML = MeetSemilattice(D) sage: P.canonical_label() Finite poset containing 3 elements sage: ML.canonical_label() Finite meet-semilattice containing 3 elements
TESTS:
sage: P = Poset(digraphs.Path(10), linear_extension=True) sage: Q = P.canonical_label() sage: Q.linear_extension() # random [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] sage: Q.cover_relations() # random [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]] sage: P = Poset(digraphs.Path(10)) sage: Q = P.canonical_label() sage: Q.linear_extension() # random [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] sage: Q.is_isomorphic(P) True sage: Poset().canonical_label() # Test the empty poset Finite poset containing 0 elements sage: D2 = Posets.DiamondPoset(4).canonical_label(algorithm='bliss') # optional: bliss sage: B2 = Posets.BooleanLattice(2).canonical_label(algorithm='bliss') # optional: bliss sage: D2 == B2 # optional: bliss True
-
cardinality
()¶ Return the number of elements in the poset.
EXAMPLES:
sage: Poset([[1,2,3],[4],[4],[4],[]]).cardinality() 5
-
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 f-vector of the poset.
Note
This is not what has been called the chain polynomial in [St1986]. The latter is identical with the order polynomial in SageMath (
order_polynomial()
).EXAMPLES:
sage: P = Posets.ChainPoset(3) sage: t = P.chain_polynomial(); t q^3 + 3*q^2 + 3*q + 1 sage: t(1) == len(list(P.chains())) True sage: P = Posets.BooleanLattice(3) sage: P.chain_polynomial() 6*q^4 + 18*q^3 + 19*q^2 + 8*q + 1 sage: P = Posets.AntichainPoset(5) sage: P.chain_polynomial() 5*q + 1
TESTS:
sage: P = Poset() sage: P.chain_polynomial() 1 sage: parent(P.chain_polynomial()) Univariate Polynomial Ring in q over Integer Ring sage: R = Poset({1: []}) sage: R.chain_polynomial() q + 1
-
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 3-dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices sage: P = posets.PentagonPoset() sage: Q = P.chain_polytope();Q A 5-dimensional polyhedron in QQ^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
TESTS:
sage: P = Poset({1: []}) sage: P.characteristic_polynomial() 1
-
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
TESTS:
sage: C = Posets.ChainPoset(10) sage: C.closed_interval(3, 3) [3] sage: C.closed_interval(8, 5) [] sage: A = Posets.AntichainPoset(10) sage: A.closed_interval(3, 7) []
-
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
TESTS:
sage: Poset().comparability_graph() Comparability graph on 0 vertices sage: C4 = Posets.ChainPoset(4) sage: C4.comparability_graph().is_isomorphic(graphs.CompleteGraph(4)) True sage: A4 = Posets.AntichainPoset(4) sage: A4.comparability_graph().is_isomorphic(Graph(4)) True
-
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 Dedekind-MacNeille completion.
See the Wikipedia article Dedekind-MacNeille completion.
OUTPUT:
- a finite lattice
EXAMPLES:
sage: P = Posets.PentagonPoset() sage: P.completion_by_cuts().is_isomorphic(P) True sage: P = Posets.AntichainPoset(3) sage: Q = P.completion_by_cuts() sage: Q.is_isomorphic(posets.DiamondPoset(5)) True sage: P = Posets.SymmetricGroupBruhatOrderPoset(3) sage: Q = P.completion_by_cuts(); Q Finite lattice containing 7 elements
See also
TESTS:
sage: Poset().completion_by_cuts() Finite lattice containing 0 elements
-
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
TESTS:
sage: Poset().connected_components() # Test empty poset [] sage: P = Poset({1: [2, 3], 3: [4, 5]}) sage: CC = P.connected_components() sage: CC[0] is P True sage: P = Poset({1: [2, 3], 3: [4, 5], 6: [7, 8]}, facade=False) sage: V = sorted(P.connected_components(), key=len)[0] sage: V(7) < V(8) # Facade argument should be inherited False
-
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
TESTS:
sage: Poset().cover_relations_graph() Graph on 0 vertices
Check that it is hashable and coincides with the Hasse diagram as a graph:
sage: P = Poset({0: [1, 2], 1: [3], 2: [3]}) sage: G = P.cover_relations_graph() sage: hash(G) == hash(G) True sage: G == Graph(P.hasse_diagram()) True
-
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()) <type '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.
See also
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
-
coxeter_transformation
()¶ Return the Coxeter transformation of the poset.
OUTPUT:
a square matrix with integer coefficients
The output is the matrix of the Auslander-Reiten translation acting on the Grothendieck group of the derived category of modules on the poset, in the basis of simple modules. This matrix is usually called the Coxeter transformation.
See also
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]
TESTS:
sage: M = posets.PentagonPoset().coxeter_transformation() sage: M ** 8 == 1 True
-
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], [], [1], [2]] sage: Pc[0] frozenset({0})
See also
REFERENCES:
[JRJ94] Jourdan, Guy-Vincent; Rampon, Jean-Xavier; Jard, Claude (1994), “Computing on-line the lattice of maximal antichains of posets”, Order 11 (3) p. 197-210, doi:10.1007/BF02115811
-
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.
See also
width()
– return the width of the poset.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]]
TESTS:
sage: p = posets.IntegerCompositions(5) sage: d = p.dilworth_decomposition() sage: for chain in d: ....: for i in range(len(chain)-1): ....: assert p.is_greater_than(chain[i],chain[i+1]) sage: set(p) == set().union(*d) True
-
dimension
(certificate=False)¶ Return the dimension of the Poset.
The (Dushnik-Miller) dimension of a poset is the minimal number of total orders so that the poset can be defined as “intersection” of all of them. Mathematically said, dimension of a poset defined on a set \(X\) of points is the smallest integer \(n\) such that there exists \(P_1,...,P_n\) linear extensions of \(P\) satisfying the following property:
\[u\leq_P v\ \text{if and only if }\ \forall i, u\leq_{P_i} v\]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.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
.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: L = P.dimension(certificate=True) sage: L # random -- architecture-dependent [[0, 2, 4, 5, 1, 3, 6], [2, 5, 0, 1, 3, 4, 6], [0, 1, 2, 3, 5, 6, 4]] sage: Poset( (L[0], lambda x, y: all(l.index(x) < l.index(y) for l in L)) ) == P True
According to Schnyder’s theorem, the poset (of height 2) of a graph has dimension \(\leq 3\) if and only if the graph is planar:
sage: G = graphs.CompleteGraph(4) sage: P = Poset(DiGraph({(u,v):[u,v] for u,v,_ in G.edges()})) sage: P.dimension() 3 sage: G = graphs.CompleteBipartiteGraph(3,3) sage: P = Poset(DiGraph({(u,v):[u,v] for u,v,_ in G.edges()})) sage: P.dimension() # not tested - around 4s with CPLEX 4
TESTS:
Empty Poset:
sage: Poset().dimension() 0 sage: Poset().dimension(certificate=True) []
REFERENCES:
[FT00] Stefan Felsner, William T. Trotter, Dimension, Graph and Hypergraph Coloring, Order, 2000, Volume 17, Issue 2, pp 167-177, http://link.springer.com/article/10.1023%2FA%3A1006429830221
-
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
TESTS:
sage: N5 = Posets.PentagonPoset() sage: P0 = Poset() sage: N5.disjoint_union(P0).is_isomorphic(N5) True sage: P0.disjoint_union(P0) Finite poset containing 0 elements sage: A3 = Posets.AntichainPoset(3) sage: A4 = Posets.AntichainPoset(4) sage: A7 = Posets.AntichainPoset(7) sage: A3.disjoint_union(A4).is_isomorphic(A7) True
-
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 meet-semilattice is join-semilattice and vice versa. Also the dual of a (non-)facade poset is again (non-)facade:
sage: V = MeetSemilattice({1: [2, 3]}, facade=False) sage: A = V.dual(); A Finite join-semilattice containing 3 elements sage: A(2) < A(1) True
See also
TESTS:
sage: Poset().dual() == Poset() # Test the empty poset True
-
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 \((\tau_1 \cdots \tau_{n-1}) (\tau_1 \cdots \tau_{n-2}) \cdots (\tau_1)\), to the default linear extension \(\pi\) ofself
(seeevacuation()
), and relabelingself
accordingly. For more details see [Stan2009].See also
linear_extension()
with_linear_extension()
and thelinear_extension
option ofPoset()
evacuation()
promotion()
REFERENCES:
[Stan2009] (1, 2, 3) Richard Stanley, Promotion and evacuation, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 9, 24 pp. 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]]
AUTHOR:
- Anne Schilling (2012-02-18)
-
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.
See also
is_bounded()
,h_polynomial()
,order_complex()
,sage.homology.cell_complex.GenericCellComplex.f_vector()
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
TESTS:
sage: P = Poset({2: []}) sage: P.f_polynomial() 1
-
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 h-vector.
See also
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
TESTS:
sage: P = Poset({2: [3]}) sage: P.flag_f_polynomial() x1 sage: P = Poset({2: []}) sage: P.flag_f_polynomial() 1
-
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
\[\prod_{k=1}^n (1-x_k) \cdot f \left(\frac{x_1}{1-x_1}, \frac{x_2}{1-x_2}, \cdots, \frac{x_n}{1-x_n}\right) \in \ZZ[x_1, x_2, \cdots, x_n],\]where \(f\) is the flag \(f\)-polynomial of \(P\) (see
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 h-vector.
See also
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
TESTS:
sage: P = Poset({2: [3]}) sage: P.flag_h_polynomial() x1 sage: P = Poset({2: []}) sage: P.flag_h_polynomial() 1
-
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 \(2|P| + 2\) vertices, defined in Section 8 of [BF1999]. Its set of vertices consists of two vertices \((0, p)\) and \((1, p)\) for each element \(p\) of \(P\), as well as two vertices \((-1, 0)\) and \((2, 0)\). (These notations are not the ones used in [BF1999]; see the table below for their relation.) The edges are:
- for each \(p\) in \(P\), an edge from \((-1, 0)\) to \((0, p)\);
- for each \(p\) in \(P\), an edge from \((1, p)\) to \((2, 0)\);
- for each \(p\) and \(q\) in \(P\) such that \(p \geq q\), an edge from \((0, p)\) to \((1, q)\).
We make this digraph into a network in the sense of flow theory as follows: The vertex \((-1, 0)\) is considered as the source of this network, and the vertex \((2, 0)\) as the sink. The cost function is defined to be \(1\) on the edge from \((0, p)\) to \((1, p)\) for each \(p \in P\), and to be \(0\) on every other edge. The capacity is \(1\) on each edge. Here is how to translate this notations into that used in [BF1999]:
our notations [BF1999] (-1, 0) s (0, p) x_p (1, p) y_p (2, 0) t a[e] a(e)
REFERENCES:
[BF1999] (1, 2, 3, 4, 5) Thomas Britz, Sergey Fomin, Finite posets and Ferrers shapes, Advances in Mathematics 158, pp. 86-127 (2001), Arxiv math/9912126 (the arXiv version has fewer errors). 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() [((-1, 0), (0, -13), None), ((-1, 0), (0, 12), None), ((-1, 0), (0, 14), None), ((-1, 0), (0, 16), None), ((0, -13), (1, -13), None), ((0, -13), (1, 12), None), ((0, -13), (1, 14), None), ((0, -13), (1, 16), None), ((0, 12), (1, 12), None), ((0, 14), (1, 12), None), ((0, 14), (1, 14), None), ((0, 16), (1, 12), None), ((0, 16), (1, 16), None), ((1, -13), (2, 0), None), ((1, 12), (2, 0), None), ((1, 14), (2, 0), None), ((1, 16), (2, 0), None)] sage: e {((-1, 0), (0, -13)): 0, ((-1, 0), (0, 12)): 0, ((-1, 0), (0, 14)): 0, ((-1, 0), (0, 16)): 0, ((0, -13), (1, -13)): 1, ((0, -13), (1, 12)): 0, ((0, -13), (1, 14)): 0, ((0, -13), (1, 16)): 0, ((0, 12), (1, 12)): 1, ((0, 14), (1, 12)): 0, ((0, 14), (1, 14)): 1, ((0, 16), (1, 12)): 0, ((0, 16), (1, 16)): 1, ((1, -13), (2, 0)): 0, ((1, 12), (2, 0)): 0, ((1, 14), (2, 0)): 0, ((1, 16), (2, 0)): 0} sage: qs = [[1,2,3,4,5,6,7,8,9],[[1,3],[3,4],[5,7],[1,9],[2,3]]] sage: Poset(qs).frank_network() (Digraph on 20 vertices, {((-1, 0), (0, 1)): 0, ((-1, 0), (0, 2)): 0, ((-1, 0), (0, 3)): 0, ((-1, 0), (0, 4)): 0, ((-1, 0), (0, 5)): 0, ((-1, 0), (0, 6)): 0, ((-1, 0), (0, 7)): 0, ((-1, 0), (0, 8)): 0, ((-1, 0), (0, 9)): 0, ((0, 1), (1, 1)): 1, ((0, 2), (1, 2)): 1, ((0, 3), (1, 1)): 0, ((0, 3), (1, 2)): 0, ((0, 3), (1, 3)): 1, ((0, 4), (1, 1)): 0, ((0, 4), (1, 2)): 0, ((0, 4), (1, 3)): 0, ((0, 4), (1, 4)): 1, ((0, 5), (1, 5)): 1, ((0, 6), (1, 6)): 1, ((0, 7), (1, 5)): 0, ((0, 7), (1, 7)): 1, ((0, 8), (1, 8)): 1, ((0, 9), (1, 1)): 0, ((0, 9), (1, 9)): 1, ((1, 1), (2, 0)): 0, ((1, 2), (2, 0)): 0, ((1, 3), (2, 0)): 0, ((1, 4), (2, 0)): 0, ((1, 5), (2, 0)): 0, ((1, 6), (2, 0)): 0, ((1, 7), (2, 0)): 0, ((1, 8), (2, 0)): 0, ((1, 9), (2, 0)): 0})
AUTHOR:
- Darij Grinberg (2013-05-09)
-
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.
REFERENCES:
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 Greene-Kleitman partition of
self
.The Greene-Kleitman partition of a finite poset \(P\) is the partition \((c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)\), where \(c_k\) is the maximum cardinality of a union of \(k\) chains of \(P\). Equivalently, this is the conjugate of the partition \((a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)\), where \(a_k\) is the maximum cardinality of a union of \(k\) antichains of \(P\).
See many sources, e. g., [BF1999], for proofs of this equivalence.
EXAMPLES:
sage: P = Poset([[3,2,1],[[3,1],[2,1]]]) sage: P.greene_shape() [2, 1] sage: P = Poset([[1,2,3,4],[[1,4],[2,4],[4,3]]]) sage: P.greene_shape() [3, 1] sage: P = Poset([[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[[1,4],[2,4],[4,3]]]) sage: P.greene_shape() [3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] sage: P = Poset([[],[]]) sage: P.greene_shape() []
AUTHOR:
- Darij Grinberg (2013-05-09)
-
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 non-facade 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:
\[h(q) = (1-q)^{\deg f} f \left( \frac{q}{1-q} \right),\]where \(f\) and \(h\) denote the \(f\)-polynomial and the \(h\)-polynomial, respectively.
See Wikipedia article h-vector.
See also
is_bounded()
,f_polynomial()
,order_complex()
,sage.homology.simplicial_complex.SimplicialComplex.h_vector()
Warning
This is slightly different from the
hPolynomial
method in Macaulay2.EXAMPLES:
sage: P = Posets.AntichainPoset(3).order_ideals_lattice() sage: P.h_polynomial() q^3 + 4*q^2 + q sage: P = Posets.DiamondPoset(5) sage: P.h_polynomial() 2*q^2 + q sage: P = Poset({1: []}) sage: P.h_polynomial() 1
-
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
TESTS:
sage: Poset().has_top() # Test empty poset False
-
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
TESTS:
sage: Poset().has_top() # Test empty poset False
-
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], [4, 12], [3, 6], [6, 12]] sage: H.edges(labels=False) [(1, 2), (1, 3), (2, 4), (2, 6), (3, 6), (4, 12), (6, 12)]
TESTS:
sage: Poset().hasse_diagram() Digraph on 0 vertices sage: P = Poset((divisors(15), attrcall("divides")), facade=True) sage: H = P.hasse_diagram() sage: H.vertices() [1, 3, 5, 15] sage: H.edges() [(1, 3, None), (1, 5, None), (3, 15, None), (5, 15, None)] sage: H.set_latex_options(format="dot2tex") # optional - dot2tex sage: view(H, tight_page=True) # optional - dot2tex, not tested (opens external window)
-
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])
TESTS:
sage: Poset().height() 0
-
incidence_algebra
(R, prefix='I')¶ Return the incidence algebra of
self
overR
.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.
See also
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
TESTS:
sage: Poset().incomparability_graph() Incomparability graph on 0 vertices sage: C4 = Posets.ChainPoset(4) sage: C4.incomparability_graph().is_isomorphic(Graph(4)) True sage: A4 = Posets.AntichainPoset(4) sage: A4.incomparability_graph().is_isomorphic(graphs.CompleteGraph(4)) True
-
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
(*args, **kwds)¶ Deprecated: Use
relations()
instead. See trac ticket #19360 for details.
-
intervals_iterator
(*args, **kwds)¶ Deprecated: Use
relations_iterator()
instead. See trac ticket #19360 for details.
-
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.
See also
EXAMPLES:
sage: P = Posets.PentagonPoset() sage: P.relations_number() 13 sage: from sage.combinat.tamari_lattices import TamariLattice sage: TamariLattice(4).relations_number() 68
TESTS:
sage: Poset().relations_number() 0
-
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 poset-morphisms from the 2-chain to \(P\).
If \(P\) is a lattice, the result is also a lattice.
EXAMPLES:
sage: P = Poset({0:[1]}) sage: P.intervals_poset() Finite poset containing 3 elements sage: P = posets.PentagonPoset() sage: P.intervals_poset() Finite lattice containing 13 elements
TESTS:
sage: P = Poset({}) sage: P.intervals_poset() Finite poset containing 0 elements sage: P = Poset({0:[]}) sage: P.intervals_poset() Finite poset containing 1 elements sage: P = Poset({0:[], 1:[]}) sage: P.intervals_poset().is_isomorphic(P) True
-
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]}
REFERENCES:
[Bj1980] Anders Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), 159-183, doi:10.1090/S0002-9947-1980-0570784-2
-
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
TESTS:
sage: Poset().is_bounded() # Test empty poset False sage: Poset({1:[]}).is_bounded() # Here top == bottom True sage: ( len([P for P in Posets(5) if P.is_bounded()]) == ....: Posets(3).cardinality() ) True
-
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
TESTS:
sage: [len([P for P in Posets(n) if P.is_chain()]) for n in range(5)] [1, 1, 1, 1, 1]
-
is_chain_of_poset
(elms, ordered=False)¶ Return
True
if \(elms\) is a chain of the poset, andFalse
otherwise.INPUT:
elms
– a list or other iterable containing some elements of the posetordered
– a Boolean. IfTrue
, then returnTrue
only if elements in \(elms\) are strictly increasing in the poset; this makes no sense if \(elms\) 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
TESTS:
sage: P = Posets.BooleanLattice(4) sage: P.is_chain_of_poset([]) True sage: P.is_chain_of_poset((1,3,7,15,14)) False sage: P.is_chain_of_poset({10}) True sage: P.is_chain_of_poset([32]) Traceback (most recent call last): ... ValueError: element (=32) not in poset
-
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
TESTS:
sage: Poset().is_connected() # Test empty poset True
-
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 non-trivial interval has the same number of elements of even rank as of odd rank. A poset is \(k\)-eulerian if every non-trivial interval up to rank \(k\) is Eulerian.
See Wikipedia 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 inteval(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 4-eulerian:
sage: P = Poset(DiGraph('MWW@_?W?@_?W??@??O@_?W?@_?W?@??O??')); P Finite poset containing 14 elements sage: P.is_eulerian(k=3) True sage: P.is_eulerian(k=4) False
Getting an interval that is not Eulerian:
sage: P = Posets.DivisorLattice(12) sage: P.is_eulerian(certificate=True) (False, (1, 4))
TESTS:
sage: Poset().is_eulerian() Traceback (most recent call last): ... TypeError: the poset is not bounded sage: Poset({1: []}).is_eulerian() True sage: Posets.PentagonPoset().is_eulerian() Traceback (most recent call last): ... TypeError: the poset is not graded sage: Posets.BooleanLattice(3).is_eulerian(k=123, certificate=True) (True, None)
-
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].
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
TESTS:
sage: Poset().is_graded() # Test empty poset True
-
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 non-facade posets also
>
works:sage: P = Poset({3: [1, 2]}, facade=False) sage: P(2) > P(3) True
See also
-
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:
- ‘’interval order’’ is \((2+2)\)-free
- ‘’semiorder’’ (or ‘’unit interval order’‘) is \((1+3)\)-free and \((2+2)\)-free
- ‘’weak order’’ is \((1+2)\)-free.
INPUT:
m
,n
- positive integers
It is also possible to give a list of integer pairs as argument. See below for an example.
EXAMPLES:
sage: B3 = Posets.BooleanLattice(3) sage: B3.is_incomparable_chain_free(1, 3) True sage: B3.is_incomparable_chain_free(2, 2) False sage: IP6 = Posets.IntegerPartitions(6) sage: IP6.is_incomparable_chain_free(1, 3) False sage: IP6.is_incomparable_chain_free(2, 2) True
A list of pairs as an argument:
sage: B3.is_incomparable_chain_free([[1, 3], [2, 2]]) False
We show how to get an incomparable chain pair:
sage: P = Posets.PentagonPoset() sage: chains_1_2 = Poset({0:[], 1:[2]}) sage: incomps = P.isomorphic_subposets(chains_1_2)[0] sage: sorted(incomps.list()), incomps.cover_relations() ([1, 2, 3], [[2, 3]])
TESTS:
sage: Poset().is_incomparable_chain_free(1,1) # Test empty poset True sage: [len([p for p in Posets(n) if p.is_incomparable_chain_free(((3, 1), (2, 2)))]) for n in range(6)] # long time [1, 1, 2, 5, 14, 42] sage: Q = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]}) sage: Q.is_incomparable_chain_free(2, 20/10) True sage: Q.is_incomparable_chain_free(2, pi) Traceback (most recent call last): ... TypeError: 2 and pi must be integers. sage: Q.is_incomparable_chain_free(2, -1) Traceback (most recent call last): ... ValueError: 2 and -1 must be positive integers. sage: P = Poset(((0, 1, 2, 3, 4), ((0, 1), (1, 2), (0, 3), (4, 2)))) sage: P.is_incomparable_chain_free((3, 1)) Traceback (most recent call last): ... TypeError: (3, 1) is not a tuple of tuples. sage: P.is_incomparable_chain_free([3, 1], [2, 2]) Traceback (most recent call last): ... TypeError: [3, 1] and [2, 2] must be integers. sage: P.is_incomparable_chain_free([[3, 1], [2, 2]]) True sage: P.is_incomparable_chain_free(([3, 1], [2, 2])) True sage: P.is_incomparable_chain_free([3, 1], 2) Traceback (most recent call last): ... TypeError: [3, 1] and 2 must be integers. sage: P.is_incomparable_chain_free(([3, 1], [2, 2, 2])) Traceback (most recent call last): ... ValueError: '([3, 1], [2, 2, 2])' is not a tuple of length-2 tuples.
AUTHOR:
- Eric Rowland (2013-05-28)
REFERENCES:
[EnumComb1] (1, 2, 3, 4, 5) Richard P. Stanley, Enumerative Combinatorics, volume 1, Second Edition, Cambridge University Press (2011). http://math.mit.edu/~rstan/ec/ec1/
-
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
TESTS:
sage: P = Poset({2:[1]}) sage: Poset().is_induced_subposet(P) True sage: Poset().is_induced_subposet(Poset()) True sage: P.is_induced_subposet(Poset()) False
Bad input:
sage: Poset().is_induced_subposet('junk') Traceback (most recent call last): ... AttributeError: 'str' object has no attribute 'subposet'
-
is_isomorphic
(other)¶ Returns 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
()¶ Return
True
if the poset has a join operation, andFalse
otherwise.A join is the least upper bound for given elements, if it exists.
EXAMPLES:
sage: P = Poset([[1,3,2], [4], [4,5,6], [6], [7], [7], [7], []]) sage: P.is_join_semilattice() True
Elements 3 and 4 have no common upper bound at all; elements 1 and 2 have upper bounds 3 and 4, but no least upper bound:
sage: P = Poset({1:[3, 4], 2:[3, 4]}) sage: P.is_join_semilattice() False
See also
TESTS:
sage: Poset().is_join_semilattice() # Test empty lattice True sage: len([P for P in Posets(4) if P.is_join_semilattice()]) 5
-
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 non-facade 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
See also
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, 3, 6, 4, 12], [1, 2, 4, 3, 6, 12], [1, 3, 2, 4, 6, 12], [1, 3, 2, 6, 4, 12]]
Note
This is used and systematically tested in
LinearExtensionsOfPosets
TESTS:
Check that trac ticket #15313 is fixed:
sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True) sage: P.is_linear_extension([1,2,4,3,6,12,1337]) False sage: P.is_linear_extension([1,2,4,3,6,666,12,1337]) False sage: P = Poset(DiGraph(5)) sage: P.is_linear_extension(['David', 'McNeil', 'La', 'Lamentable', 'Aventure', 'de', 'Simon', 'Wiesenthal']) False
-
is_meet_semilattice
()¶ Return
True
if the poset has a meet operation, andFalse
otherwise.A meet is the greatest lower bound for given elements, if it exists.
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
See also
TESTS:
sage: Poset().is_meet_semilattice() # Test empty lattice True sage: len([P for P in Posets(4) if P.is_meet_semilattice()]) 5
-
is_parent_of
(x)¶ Returns True if x is an element of the poset.
TESTS:
sage: from sage.combinat.posets.posets import FinitePoset sage: P5 = FinitePoset(DiGraph({(5,):[(4,1),(3,2)], \ (4,1):[(3,1,1),(2,2,1)], \ (3,2):[(3,1,1),(2,2,1)], \ (3,1,1):[(2,1,1,1)], \ (2,2,1):[(2,1,1,1)], \ (2,1,1,1):[(1,1,1,1,1)], \ (1,1,1,1,1):[]})) sage: x = P5.list()[3] sage: x in P5 True
For the sake of speed, an element with the right class and parent is assumed to be in this parent. This can possibly be counterfeited by feeding garbage to the constructor:
sage: x = P5.element_class(P5, "a", 5) sage: x in P5 True
-
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 \(h-i\) for every \(i\) in \(0, 1, \ldots, h\).
EXAMPLES:
sage: P = Poset({1:[3, 4, 5], 2:[3, 4, 5], 3:[6], 4:[7], 5:[7]}) sage: P.is_rank_symmetric() True sage: P = Poset({1:[2], 2:[3, 4], 3:[5], 4:[5]}) sage: P.is_rank_symmetric() False
TESTS:
sage: Poset().is_rank_symmetric() # Test empty poset True
-
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
TESTS:
sage: Poset().is_ranked() # Test empty poset True
-
is_series_parallel
()¶ Return
True
if the poset is series-parallel, andFalse
otherwise.A poset is series-parallel if it can be built up from one-element posets using the operations of disjoint union and ordinal sum. This is also called N-free property: every poset that is not series-parallel contains a subposet isomorphic to the 4-element N-shaped poset where \(a > c, d\) and \(b > d\).
See Wikipedia article Series-parallel partial order.
EXAMPLES:
sage: VA = Poset({1: [2, 3], 4: [5], 6: [5]}) sage: VA.is_series_parallel() True sage: big_N = Poset({1: [2, 4], 2: [3], 4:[7], 5:[6], 6:[7]}) sage: big_N.is_series_parallel() False
TESTS:
sage: Poset().is_series_parallel() True
-
is_slender
()¶ Return
True
if the poset is slender, andFalse
otherwise.A finite graded poset is called 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.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
TESTS:
sage: Poset().is_slender() # Test empty poset True
-
isomorphic_subposets
(other)¶ Return a list of subposets of \(self\) isomorphic to \(other\).
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: for x in C3.isomorphic_subposets(C2): ....: print(x.cover_relations()) [['b', 'c']] [['a', 'c']] [['a', 'b']] 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 to \(other\).
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.
-
join_matrix
()¶ Deprecated as a function of posets, moved to lattices.
Convert a poset \(P\) to join-semilattice and use it like
JoinSemilattice(P).join_matrix()
.
-
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])
REFERENCES:
[BIANCO] L. Bianco, P. Dell‘Olmo, S. Giordani An Optimal Algorithm to Find the Jump Number of Partially Ordered Sets Computational Optimization and Applications, 1997, Volume 8, Issue 2, pp 197–210, doi:10.1023/A:1008625405476 TESTS:
sage: E = Poset() sage: E.jump_number(certificate=True) (0, []) sage: C4 = Posets.ChainPoset(4) sage: A4 = Posets.AntichainPoset(4) sage: C4.jump_number() 0 sage: A4.jump_number() 3
-
kazhdan_lusztig_polynomial
(x=None, y=None, q=None, canonical_labels=None)¶ Return the Kazhdan-Lusztig polynomial \(P_{x,y}(q)\) of the poset.
The poset is expected to be ranked.
We follow the definition given in [EPW14]. Let \(G\) denote a graded poset with unique minimal and maximal elements and \(\chi_G\) denote the characteristic polynomial of \(G\). Let \(I_x\) and \(F^x\) denote the principal order ideal and filter of \(x\) respectively. Define the Kazhdan-Lusztig polynomial of \(G\) as the unique polynomial \(P_G(q)\) satisfying the following:
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
REFERENCES:
[EPW14] Ben Elias, Nicholas Proudfoot, and Max Wakefield. The Kazhdan-Lusztig polynomial of a matroid. 2014. Arxiv 1412.7408. AUTHORS:
- Travis Scrimshaw (27-12-2014)
-
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)¶ Computes 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 a list
l
such thatl[i]
is the set of minimal elements of the poset obtained fromself
by removing the elements inl[0], l[1], ..., l[i-1]
. (In particular,l[0]
is the set of minimal elements ofself
.)EXAMPLES:
sage: P = Poset({0:[1,2],1:[3],2:[3],3:[]}) sage: [len(x) for x in P.level_sets()] [1, 2, 1]
sage: Q = Poset({0:[1,2], 1:[3], 2:[4], 3:[4]}) sage: [len(x) for x in Q.level_sets()] [1, 2, 1, 1]
-
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 \(\pi := \pi_0 \pi_1 \ldots \pi_{n-1}\) of its elements such that \(i<j\) whenever \(\pi_i < \pi_j\) in the poset \(P\).
INPUT:
linear_extension
– (default:None
) a list of the elements ofself
check
– a boolean (default: True); whether to check thatlinear_extension
is indeed a linear extension ofself
.
See also
EXAMPLES:
sage: P = Poset((divisors(15), attrcall("divides")), facade=True)
Without optional argument, the default linear extension of the poset is returned, as a plain list:
sage: P.linear_extension() [1, 3, 5, 15]
Otherwise, a full-featured linear extension is constructed as an element of
P.linear_extensions()
:sage: l = P.linear_extension([1,5,3,15]); l [1, 5, 3, 15] sage: type(l) <class 'sage.combinat.posets.linear_extensions.LinearExtensionsOfPoset_with_category.element_class'> sage: l.parent() The set of all linear extensions of Finite poset containing 4 elements
By default, the linear extension is checked for correctness:
sage: l = P.linear_extension([1,3,15,5]) Traceback (most recent call last): ... ValueError: [1, 3, 15, 5] is not a linear extension of Finite poset containing 4 elements
This can be disabled (at your own risks!) with:
sage: P.linear_extension([1,3,15,5], check=False) [1, 3, 15, 5]
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
See also
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, 3, 6, 4, 12], [1, 2, 4, 3, 6, 12], [1, 3, 2, 4, 6, 12], [1, 3, 2, 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) <type '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, 3, 6, 4, 12], [1, 2, 4, 3, 6, 12], [1, 3, 2, 4, 6, 12], [1, 3, 2, 6, 4, 12]] sage: type(L[0]) <type 'list'>
TESTS:
sage: D = Poset({ 0:[1,2], 1:[3], 2:[3,4] }) sage: list(D.linear_extensions()) [[0, 1, 2, 3, 4], [0, 1, 2, 4, 3], [0, 2, 1, 3, 4], [0, 2, 1, 4, 3], [0, 2, 4, 1, 3]]
-
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, 2, 4, 3], [1, 3, 2, 4]] sage: chevron = Poset({1: [2, 6], 2: [3], 4: [3, 5], 6: [5]}) sage: G = chevron.linear_extensions_graph(); G Graph on 22 vertices sage: G.size() 36
TESTS:
sage: Poset().linear_extensions_graph() Graph on 1 vertex sage: A4 = Posets.AntichainPoset(4) sage: G = A4.linear_extensions_graph() sage: G.is_regular() True
-
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.elements.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) <type '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 non-facade 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: 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
-
meet_matrix
()¶ Deprecated as a function of posets, moved to lattices.
Convert a poset \(P\) to meet-semilattice and use it like
MeetSemilattice(P).join_matrix()
.
-
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
-
mobius_function
(*args, **kwds)¶ Deprecated: Use
moebius_function()
instead. See trac ticket #19855 for details.
-
mobius_function_matrix
(*args, **kwds)¶ Deprecated: Use
moebius_function_matrix()
instead. See trac ticket #19855 for details.
-
moebius_function
(x, y)¶ Returns 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)¶ Returns 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
TESTS:
sage: C = Posets.ChainPoset(10) sage: C.open_interval(3, 3) [] sage: C.open_interval(3, 4) [] sage: C.open_interval(7, 3) [] sage: A = Posets.AntichainPoset(10) sage: A.open_interval(3, 7) []
-
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, 2, 3), (0, 1, 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
TESTS:
sage: P = Poset() # Test empty poset sage: P.order_filter([]) [] sage: C = Posets.ChainPoset(5) sage: C.order_filter([]) []
-
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
TESTS:
sage: P = Poset() # Test empty poset sage: P.order_ideal([]) [] sage: C = Posets.ChainPoset(5) sage: C.order_ideal([]) []
-
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 order-preserving maps from \(P\) to \(\{1,\ldots,m\}\).
See sections 3.12 and 3.15 of [EnumComb1], and also [St1986].
See also
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]
-
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 3-dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices sage: P = posets.PentagonPoset() sage: Q = P.order_polytope();Q A 5-dimensional polyhedron in QQ^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
REFERENCES:
[St1986] (1, 2, 3, 4) Richard Stanley. Two poset polytopes, Discrete Comput. Geom. (1986), doi:10.1007/BF02187680
-
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
TESTS:
sage: P1.ordinal_product(24) Traceback (most recent call last): ... ValueError: the input is not a finite poset sage: P1.ordinal_product(P2, labels='camembert') Traceback (most recent call last): ... ValueError: labels must be either 'pairs' or 'integers' sage: N5 = Posets.PentagonPoset() sage: P0 = Poset() sage: N5.ordinal_product(P0) == P0 True sage: P0.ordinal_product(N5) == P0 True sage: P0.ordinal_product(P0) == P0 True sage: A3 = Posets.AntichainPoset(3) sage: A4 = Posets.AntichainPoset(4) sage: A12 = Posets.AntichainPoset(12) sage: A3.ordinal_product(A4).is_isomorphic(A12) True sage: C3 = Posets.ChainPoset(3) sage: C4 = Posets.ChainPoset(4) sage: C12 = Posets.ChainPoset(12) sage: C3.ordinal_product(C4).is_isomorphic(C12) True
-
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 join-semilattices.
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 join-semilattice containing 2 elements sage: L = LatticePoset({1:[2]}); L Finite lattice containing 2 elements sage: P.ordinal_sum(L) Finite poset containing 4 elements sage: L.ordinal_sum(JL) Finite join-semilattice containing 4 elements sage: L.ordinal_sum(L) Finite lattice containing 4 elements
See also
TESTS:
sage: N5 = Posets.PentagonPoset() sage: P0 = LatticePoset({}) sage: N5.ordinal_sum(P0).is_isomorphic(N5) True sage: P0.ordinal_sum(P0) Finite lattice containing 0 elements
-
ordinal_summands
()¶ Return the ordinal summands of the poset as subposets.
The ordinal summands of a poset \(P\) is the longest list of non-empty subposets \(P_1, \ldots, P_n\) whose ordinal sum is \(P\). This decomposition is unique.
EXAMPLES:
sage: P = Poset({'a': ['c', 'd'], 'b': ['d'], 'c': ['x', 'y'], ....: 'd': ['x', 'y']}) sage: parts = P.ordinal_summands(); parts [Finite poset containing 4 elements, Finite poset containing 2 elements] sage: sorted(parts[0]) ['a', 'b', 'c', 'd'] sage: Q = parts[0].ordinal_sum(parts[1]) sage: Q.is_isomorphic(P) True
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.
TESTS:
sage: Poset().ordinal_summands() [Finite poset containing 0 elements] sage: Poset({1: []}).ordinal_summands() [Finite poset containing 1 elements]
-
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]
REFERENCES:
[Grinb2016a] Darij Grinberg, Double posets and the antipode of QSym, Arxiv 1509.08355v2.
-
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:
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.layout
– the type of layout used to display the Diagram. Set to'acyclic'
by default (seeGenericGraph.plot
for more information).
Note
All options of
GenericGraph.plot
are also available through this function.EXAMPLES:
sage: D = Poset({ 1:[2,3], 2:[4], 3:[4,5] }) sage: D.plot(label_elements=False) Graphics object consisting of 6 graphics primitives sage: D.plot() Graphics object consisting of 11 graphics primitives sage: type(D.plot()) <class 'sage.plot.graphics.Graphics'> sage: elm_labs = {1:'a', 2:'b', 3:'c', 4:'d', 5:'e'} sage: D.plot(element_labels=elm_labs) Graphics object consisting of 11 graphics primitives
Plot of a ranked poset:
sage: P = Poset(DiGraph('E@ACA@?')) sage: P.is_ranked() True sage: P.plot() Graphics object consisting of 12 graphics primitives
The keyword
cover_labels
can be used to decorate edges:sage: P = posets.ChainPoset(3) sage: P.plot(cover_labels=lambda a, b: a + b) Graphics object consisting of 8 graphics primitives sage: P = Poset({0: [1,2]}) sage: P.plot(cover_labels={(0,1): 'here', (0,2): 'there'}) Graphics object consisting of 8 graphics primitives sage: P = Poset({2: [1], 0: [1]}) sage: P.plot(cover_labels=[(2,1,'da'), (0,1,'niet')]) Graphics object consisting of 8 graphics primitives
TESTS:
We check that
label_elements
andelement_labels
are honored:sage: def get_plot_labels(P): return sorted(t.string for t in P if isinstance(t, sage.plot.text.Text)) sage: P1 = Poset({ 0:[1,2], 1:[3], 2:[3,4] }) sage: P2 = Poset({ 0:[1,2], 1:[3], 2:[3,4] }, facade=True) sage: get_plot_labels(P1.plot(label_elements=False)) [] sage: get_plot_labels(P1.plot(label_elements=True)) [u'0', u'1', u'2', u'3', u'4'] sage: element_labels = {0:'a', 1:'b', 2:'c', 3:'d', 4:'e'} sage: get_plot_labels(P1.plot(element_labels=element_labels)) [u'a', u'b', u'c', u'd', u'e'] sage: get_plot_labels(P2.plot(element_labels=element_labels)) [u'a', u'b', u'c', u'd', u'e']
The following checks that trac ticket #18936 has been fixed and labels still work:
sage: P = Poset({0: [1,2], 1:[3]}) sage: heights = {1 : [0], 2 : [1], 3 : [2,3]} sage: P.plot(heights=heights) Graphics object consisting of 8 graphics primitives sage: elem_labels = {0 : 'a', 1 : 'b', 2 : 'c', 3 : 'd'} sage: P.plot(element_labels=elem_labels, heights=heights) Graphics object consisting of 8 graphics primitives
Plot of the empy poset:
sage: P = Poset({}) sage: P.plot() Graphics object consisting of 0 graphics primitives
-
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
TESTS:
sage: Poset({0: [1]}).product(Poset()) # Product with empty poset Finite poset containing 0 elements sage: Poset().product(Poset()) # Product of two empty poset Finite poset containing 0 elements
We check that trac ticket #19113 is fixed:
sage: L = LatticePoset({1: []}) sage: type(L) == type(L.product(L)) True
-
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 \(\tau_i \tau_{i+1} \cdots \tau_{n-1}\) 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 (2012-02-18)
-
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
REFERENCES:
[Propp1997] James Propp, Generating Random Elements of Finite Distributive Lattices, Electron. J. Combin. 4 (1997), no. 2, The Wilf Festschrift volume, Research Paper 15. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r15
-
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]]
TESTS:
sage: P = Posets.IntegerPartitions(4) sage: P.random_subposet(1) == P True
-
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.)
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
TESTS:
sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]]) sage: r = P.rank_function() sage: for u,v in P.cover_relations_iterator(): ....: if r(v) != r(u) + 1: ....: print("Bug in rank_function!")
sage: Q = Poset([[1,2],[4],[3],[4],[]]) sage: Q.rank_function() is None True
-
relabel
(relabeling)¶ Return a copy of this poset with its elements relabelled.
INPUT:
relabeling
– a function or dictionary
This function must map each (non-wrapped) element of
self
to some distinct object.EXAMPLES:
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]]
Here we relabel the elements of a poset by \(\{0,1,2, ...\}\), 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)}; 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:sage: P = Poset((divisors(12), attrcall("divides")), facade = True, linear_extension=True) sage: P.list() [1, 2, 3, 4, 6, 12] sage: Q = P.relabel(lambda x: 12/x) sage: Q.list() [12, 6, 4, 3, 2, 1] sage: Q.cover_relations() [[12, 6], [12, 4], [6, 3], [6, 2], [4, 2], [3, 1], [2, 1]]
Relabeling a (semi)lattice gives a (semi)lattice:
sage: P = JoinSemilattice({0: [1]}) sage: P.relabel(lambda n: n+1) Finite join-semilattice containing 2 elements
Note
As can be seen in the above examples, the default linear extension of
Q
is that ofP
after relabeling. In particular,P
andQ
share the same internal Hasse diagram.TESTS:
Test non-facade poset:
sage: P = Poset({3: [2]}, facade=False) sage: Q = P.relabel(lambda x: chr(ord('a')+x)) sage: Q('c') < Q('d') False
The following checks that trac ticket #14019 has been fixed:
sage: d = DiGraph({2:[1],3:[1]}) sage: p1 = Poset(d) sage: p2 = p1.relabel({1:1,2:3,3:2}) sage: p1.hasse_diagram() == p2.hasse_diagram() True sage: p1 == p2 True sage: d = DiGraph({2:[1],3:[1]}) sage: p1 = Poset(d) sage: p2 = p1.relabel({1:2,2:3,3:1}) sage: p3 = p2.relabel({2:1,1:2,3:3}) sage: p1.hasse_diagram() == p3.hasse_diagram() True sage: p1 == p3 True
-
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.
See also
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]]
TESTS:
sage: P = Poset() # Test empty poset sage: P.relations() []
AUTHOR:
- Rob Beezer (2011-05-04)
-
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.
See also
EXAMPLES:
sage: P = Poset({0:[2], 1:[2], 2:[3], 3:[4], 4:[]}) sage: it = P.relations_iterator() sage: type(it) <type '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]]
AUTHOR:
- Rob Beezer (2011-05-04)
-
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.
See also
EXAMPLES:
sage: P = Posets.PentagonPoset() sage: P.relations_number() 13 sage: from sage.combinat.tamari_lattices import TamariLattice sage: TamariLattice(4).relations_number() 68
TESTS:
sage: Poset().relations_number() 0
-
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
TESTS:
sage: P = Posets.PentagonPoset() sage: P.sorted([], allow_incomparable=True, remove_duplicates=True) [] sage: P.sorted([], allow_incomparable=False, remove_duplicates=True) [] sage: P.sorted([], allow_incomparable=True, remove_duplicates=False) [] sage: P.sorted([], allow_incomparable=False, remove_duplicates=False) []
-
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: ABC.star_product(XYZ, labels='integers').list() [0, 1, 2, 3]
TESTS:
sage: C0 = Poset() sage: C1 = Poset({0: []}) sage: C2 = Poset({0: [1]}) sage: C2.star_product(42) Traceback (most recent call last): ... TypeError: the input is not a finite poset sage: C2.star_product(C0) Traceback (most recent call last): ... ValueError: the poset and 'other' should be bounded sage: C0.star_product(C2) Traceback (most recent call last): ... ValueError: the poset and 'other' should be bounded sage: C2.star_product(C1) Traceback (most recent call last): ... ValueError: the poset and 'other' should have at least two elements sage: C1.star_product(C2) Traceback (most recent call last): ... ValueError: the poset and 'other' should have at least two elements
-
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 non-facade poset is again a non-facade poset:
sage: P = Posets.PentagonPoset(facade=False) sage: Q = P.subposet([0, 1, 2, 4]) sage: Q(1) < Q(2) False
TESTS:
sage: P = Poset({'a': ['b'], 'b': ['c']}) sage: P.subposet(('a', 'b', 'c')) Finite poset containing 3 elements sage: P.subposet([]) Finite poset containing 0 elements sage: P.subposet(["a","b","x"]) Traceback (most recent call last): ... ValueError: <type 'str'> is not an element of this poset sage: P.subposet(3) Traceback (most recent call last): ... TypeError: 'sage.rings.integer.Integer' object is not iterable
-
to_graph
(*args, **kwds)¶ Deprecated: Use
cover_relations_graph()
instead. See trac ticket #17449 for details.
-
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
TESTS:
sage: R = Poset([[0],[]]) sage: R.list() [0] sage: R.top() #Trac #10776 0
-
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)) <type '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]
TESTS:
sage: Poset().width() 0 sage: Poset().width(certificate=True) (0, [])
-
with_bounds
(labels=('bottom', 'top'))¶ Return the poset with bottom and top elements adjoined.
This functions always adds two new elements to the poset, i.e. 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
TESTS:
sage: P = Poset().with_bounds() sage: P.cover_relations() [['bottom', 'top']] sage: L = LatticePoset({}).with_bounds(); L Finite lattice containing 2 elements sage: L.meet_irreducibles() # Trac 21543 ['bottom'] sage: Poset().with_bounds((None, 1)) Finite poset containing 1 elements sage: LatticePoset().with_bounds((None, 1)) Finite lattice containing 1 elements sage: MeetSemilattice().with_bounds((None, 1)) Finite lattice containing 1 elements sage: JoinSemilattice().with_bounds((None, 1)) Finite join-semilattice containing 1 elements sage: Poset().with_bounds((1, None)) Finite poset containing 1 elements sage: LatticePoset().with_bounds((1, None)) Finite lattice containing 1 elements sage: MeetSemilattice().with_bounds((1, None)) Finite meet-semilattice containing 1 elements sage: JoinSemilattice().with_bounds((1, None)) Finite lattice containing 1 elements sage: P = Poset({0: []}) sage: L = LatticePoset({0: []}) sage: ML = MeetSemilattice({0: []}) sage: JL = JoinSemilattice({0: []}) sage: P.with_bounds((None, None)) Finite poset containing 1 elements sage: L.with_bounds((None, None)) Finite lattice containing 1 elements sage: ML.with_bounds((None, None)) Finite meet-semilattice containing 1 elements sage: JL.with_bounds((None, None)) Finite join-semilattice containing 1 elements sage: P.with_bounds((1, None)) Finite poset containing 2 elements sage: L.with_bounds((1, None)) Finite lattice containing 2 elements sage: ML.with_bounds((1, None)) Finite meet-semilattice containing 2 elements sage: JL.with_bounds((1, None)) Finite lattice containing 2 elements sage: P.with_bounds((None, 1)) Finite poset containing 2 elements sage: L.with_bounds((None, 1)) Finite lattice containing 2 elements sage: ML.with_bounds((None, 1)) Finite lattice containing 2 elements sage: JL.with_bounds((None, 1)) Finite join-semilattice containing 2 elements sage: Posets.PentagonPoset().with_bounds(labels=(4, 5)) Traceback (most recent call last): ... ValueError: the poset already has element 4
-
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]]
TESTS:
We check that we can pass in a list of elements of
P
instead:sage: Q = P.with_linear_extension([P(_) for _ in [1,3,2,6,4,12]]) sage: list(Q) [1, 3, 2, 6, 4, 12] sage: Q.cover_relations() [[1, 3], [1, 2], [3, 6], [2, 6], [2, 4], [6, 12], [4, 12]]
We check that this works for facade posets too:
sage: P = Poset((divisors(12), attrcall("divides")), facade=True) sage: Q = P.with_linear_extension([1,3,2,6,4,12]) sage: list(Q) [1, 3, 2, 6, 4, 12] sage: Q.cover_relations() [[1, 3], [1, 2], [3, 6], [2, 6], [2, 4], [6, 12], [4, 12]] sage: sorted(Q.cover_relations()) == sorted(P.cover_relations()) True
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.
-
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 \(x_1 \leq x_2 \leq \dots \leq x_{m-1}\) of elements of the poset.
The polynomial \(Z(q)\) is integral-valued, but generally doesn’t have integer coefficients. It can be computed as
\[Z(q) = \sum_{k \geq 1} \dbinom{q-2}{k-1} c_k,\]where \(c_k\) is the number of all chains of length \(k\) in the poset.
For more information, see section 3.12 of [EnumComb1].
In particular, \(Z(2)\) is the number of vertices and \(Z(3)\) is the number of intervals.
EXAMPLES:
sage: Posets.ChainPoset(2).zeta_polynomial() q sage: Posets.ChainPoset(3).zeta_polynomial() 1/2*q^2 + 1/2*q sage: P = posets.PentagonPoset() sage: P.zeta_polynomial() 1/6*q^3 + q^2 - 1/6*q sage: P = Posets.DiamondPoset(5) sage: P.zeta_polynomial() 3/2*q^2 - 1/2*q
TESTS:
Checking the simplest cases:
sage: Poset({}).zeta_polynomial() 0 sage: Poset({1: []}).zeta_polynomial() 1 sage: Poset({1: [], 2: []}).zeta_polynomial() 2
-
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 pre-computed values obtained from the On-Line 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 two-element 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 two-element 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 two-element 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, list or tuple of upper covers:
data[x]
is a list of the elements that cover the element \(x\) in the poset.Warning
If data is a list or tuple of length \(2\), then it is handled by the above case..
An acyclic, loop-free and multi-edge 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 non-cover relations:
sage: elms = [1,2,3,4] sage: rels = [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]] sage: P = Poset( [elms,rels] ,cover_relations=False); P Finite poset containing 4 elements sage: P.cover_relations() [[1, 2], [2, 3], [3, 4]]
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:
sage: Poset([[1,2],[4],[3],[4],[]]) Finite poset containing 5 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 above cases. So you cannot use this method to input a 2-element 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) <type '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]) <type '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) <type 'sage.rings.integer.Integer'> sage: P = Poset((divisors(15), attrcall("divides")), facade=True) sage: type(P.an_element()) <type '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.TESTS:
sage: P = Poset([[1,2],[3],[3]]) sage: type(hash(P)) <type 'int'>
Bad input:
sage: Poset([1,2,3], lambda x,y : x<y) Traceback (most recent call last): ... ValueError: element_labels should be a dict or a list if different from None. (Did you intend data to be equal to a pair ?)
Another kind of bad input, digraphs with oriented cycles:
sage: Poset(DiGraph([[1,2],[2,3],[3,4],[4,1]])) Traceback (most recent call last): ... ValueError: The graph is not directed acyclic
-
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