Poset -- a class for partially ordered sets (posets)
Description
This class is a type of HashTable which represents finite posets. It consists of a ground set, a list of relationships {a,b} where a ≤b, and a matrix encoding these relations.
i1 : G = {1,2,3,4}; -- the ground set
i2 : R = {{1,2},{1,3},{2,4},{3,4}}; -- a list of cover relations
i3 : P = poset(G, R) -- the poset with its relations matrix computed
o3 = P
o3 : Poset
projectivizeArrangement(List,Ring), see projectivizeArrangement -- computes the intersection poset of a projectivized hyperplane arrangement
randomPoset(List), see randomPoset -- generates a random poset with a given relation probability
randomPoset(ZZ), see randomPoset -- generates a random poset with a given relation probability
resolutionPoset(ChainComplex), see resolutionPoset -- generates a poset from a resolution
resolutionPoset(Ideal), see resolutionPoset -- generates a poset from a resolution
resolutionPoset(MonomialIdeal), see resolutionPoset -- generates a poset from a resolution
standardMonomialPoset(MonomialIdeal), see standardMonomialPoset -- generates the poset of divisibility in the monomial basis of an ideal
standardMonomialPoset(MonomialIdeal,ZZ,ZZ), see standardMonomialPoset -- generates the poset of divisibility in the monomial basis of an ideal
subposet(Poset,List), see subposet -- computes the induced subposet of a poset given a list of elements
transitiveOrientation(Graph), see transitiveOrientation -- generates a poset whose comparability graph is the given graph
union(Poset,Poset), see union -- computes the union of two posets
youngSubposet(List), see youngSubposet -- generates a subposet of Young's lattice
youngSubposet(List,List), see youngSubposet -- generates a subposet of Young's lattice
youngSubposet(ZZ), see youngSubposet -- generates a subposet of Young's lattice
Methods that use an object of class Poset :
allRelations(Poset), see allRelations -- computes all relations of a poset
allRelations(Poset,Boolean), see allRelations -- computes all relations of a poset
antichains(Poset), see antichains -- computes all antichains of a poset
antichains(Poset,ZZ), see antichains -- computes all antichains of a poset
areIsomorphic(Poset,Poset), see areIsomorphic -- determines if two posets are isomorphic
Poset == Poset, see areIsomorphic -- determines if two posets are isomorphic
atoms(Poset), see atoms -- computes the list of elements covering the minimal elements of a poset
chains(Poset), see chains -- computes all chains of a poset
chains(Poset,ZZ), see chains -- computes all chains of a poset
characteristicPolynomial(Poset), see characteristicPolynomial -- computes the characteristic polynomial of a ranked poset with a unique minimal element
comparabilityGraph(Poset), see comparabilityGraph -- produces the comparability graph of a poset
compare(Poset,Thing,Thing), see compare -- compares two elements in a poset