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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 $a \leq b$, 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

See also

Functions and methods returning an object of class Poset:

  • adjoinMax(Poset) -- see adjoinMax -- computes the poset with a new maximum element
  • adjoinMax(Poset,Thing) -- see adjoinMax -- computes the poset with a new maximum element
  • adjoinMin(Poset) -- see adjoinMin -- computes the poset with a new minimum element
  • adjoinMin(Poset,Thing) -- see adjoinMin -- computes the poset with a new minimum element
  • augmentPoset(Poset) -- see augmentPoset -- computes the poset with an adjoined minimum and maximum
  • augmentPoset(Poset,Thing,Thing) -- see augmentPoset -- computes the poset with an adjoined minimum and maximum
  • booleanLattice(ZZ) -- see booleanLattice -- generates the boolean lattice on $n$ elements
  • chain(ZZ) -- see chain -- generates the chain poset on $n$ elements
  • closedInterval(Poset,Thing,Thing) -- see closedInterval -- computes the subposet contained between two points
  • diamondProduct(Poset,Poset) -- see diamondProduct -- computes the diamond product of two ranked posets
  • dilworthLattice(Poset) -- see dilworthLattice -- computes the Dilworth lattice of a poset
  • distributiveLattice(Poset) -- see distributiveLattice -- computes the lattice of order ideals of a poset
  • divisorPoset(ZZ) -- see divisorPoset -- generates the poset of divisors
  • divisorPoset(List,List,PolynomialRing) -- generates the poset of divisors
  • divisorPoset(RingElement) -- generates the poset of divisors
  • divisorPoset(RingElement,RingElement) -- generates the poset of divisors with a lower and upper bound
  • dominanceLattice(ZZ) -- see dominanceLattice -- generates the dominance lattice of partitions of $n$
  • dropElements(Poset,Function) -- see dropElements -- computes the induced subposet of a poset given a list of elements to remove
  • dropElements(Poset,List) -- see dropElements -- computes the induced subposet of a poset given a list of elements to remove
  • dual(Poset) -- produces the derived poset with relations reversed
  • facePoset(SimplicialComplex) -- see facePoset -- generates the face poset of a simplicial complex
  • flagPoset(Poset,List) -- see flagPoset -- computes the subposet of specified ranks of a ranked poset
  • gapConvertPoset(Array) -- see gapConvertPoset -- converts between Macaulay2's Posets and GAP's Posets
  • gapConvertPoset(String) -- see gapConvertPoset -- converts between Macaulay2's Posets and GAP's Posets
  • indexLabeling(Poset) -- see indexLabeling -- relabels a poset with the labeling based on the indices of the vertices
  • intersectionLattice(List,Ring) -- see intersectionLattice -- generates the intersection lattice of a hyperplane arrangement
  • labelPoset(Poset,HashTable) -- see labelPoset -- relabels a poset with the specified labeling
  • lcmLattice(Ideal) -- see lcmLattice -- generates the lattice of lcms in an ideal
  • naturalLabeling(Poset) -- see naturalLabeling -- relabels a poset with a natural labeling
  • naturalLabeling(Poset,ZZ) -- see naturalLabeling -- relabels a poset with a natural labeling
  • ncpLattice(ZZ) -- see ncpLattice -- computes the non-crossing partition lattice of set-partitions of size $n$
  • openInterval(Poset,Thing,Thing) -- see openInterval -- computes the subposet contained strictly between two points
  • partitionLattice(ZZ) -- see partitionLattice -- computes the lattice of set-partitions of size $n$
  • plueckerPoset(ZZ) -- see plueckerPoset -- computes a poset associated to the Plücker relations
  • poset(List) -- see poset -- creates a new Poset object
  • poset(List,Function) -- see poset -- creates a new Poset object
  • poset(List,List) -- see poset -- creates a new Poset object
  • poset(List,List,Matrix) -- see poset -- creates a new Poset object
  • product(Poset,Poset) -- computes the product of two posets
  • 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(Complex) -- 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
  • Poset + Poset -- see union(Poset,Poset) -- computes the union of two posets
  • union(Poset,Poset) -- 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
  • connectedComponents(Poset) -- generates a list of connected components of a poset
  • coveringRelations(Poset) -- see coveringRelations -- computes the minimal list of generating relations of a poset
  • coxeterPolynomial(Poset) -- see coxeterPolynomial -- computes the Coxeter polynomial of a poset
  • degreePolynomial(Poset) -- see degreePolynomial -- computes the degree polynomial of a poset
  • dilworthNumber(Poset) -- see dilworthNumber -- computes the Dilworth number of a poset
  • displayPoset(Poset) -- see displayPoset -- generates a PDF representation of a poset and attempts to display it
  • Poset - List -- see dropElements -- computes the induced subposet of a poset given a list of elements to remove
  • filter(Poset,List) -- see filter -- computes the elements above given elements in a poset
  • filtration(Poset) -- see filtration -- generates the filtration of a poset
  • flagChains(Poset,List) -- see flagChains -- computes the maximal chains in a list of flags of a ranked poset
  • flagfPolynomial(Poset) -- see flagfPolynomial -- computes the flag-f polynomial of a ranked poset
  • flaghPolynomial(Poset) -- see flaghPolynomial -- computes the flag-h polynomial of a ranked poset
  • fPolynomial(Poset) -- see fPolynomial -- computes the f-polynomial of a poset
  • gapConvertPoset(Poset) -- see gapConvertPoset -- converts between Macaulay2's Posets and GAP's Posets
  • greeneKleitmanPartition(Poset) -- see greeneKleitmanPartition -- computes the Greene-Kleitman partition of a poset
  • hasseDiagram(Poset) -- see hasseDiagram -- produces the Hasse diagram of a poset
  • height(Poset) -- computes the height of a poset
  • hibiIdeal(Poset) -- see hibiIdeal -- produces the Hibi ideal of a poset
  • hibiRing(Poset) -- see hibiRing -- produces the Hibi ring of a poset
  • hPolynomial(Poset) -- see hPolynomial -- computes the h-polynomial of a poset
  • incomparabilityGraph(Poset) -- see incomparabilityGraph -- produces the incomparability graph of a poset
  • isAntichain(Poset,List) -- see isAntichain -- determines if a given list of vertices is an antichain of a poset
  • isAtomic(Poset) -- see isAtomic -- determines if a lattice is atomic
  • isBounded(Poset) -- see isBounded -- determines if a poset is bounded
  • isConnected(Poset) -- determines if a poset is connected
  • isDistributive(Poset) -- see isDistributive -- determines if a lattice is distributive
  • isEulerian(Poset) -- determines if a ranked poset is Eulerian
  • isGeometric(Poset) -- see isGeometric -- determines if a lattice is geometric
  • isGraded(Poset) -- see isGraded -- determines if a poset is graded
  • isLattice(Poset) -- see isLattice -- determines if a poset is a lattice
  • isLowerSemilattice(Poset) -- see isLowerSemilattice -- determines if a poset is a lower (or meet) semilattice
  • isLowerSemimodular(Poset) -- see isLowerSemimodular -- determines if a ranked lattice is lower semimodular
  • isModular(Poset) -- see isModular -- determines if a lattice is modular
  • isomorphism(Poset,Poset) -- computes an isomorphism between isomorphic posets
  • isRanked(Poset) -- see isRanked -- determines if a poset is ranked
  • isSperner(Poset) -- see isSperner -- determines if a ranked poset has the Sperner property
  • isStrictSperner(Poset) -- see isStrictSperner -- determines if a ranked poset has the strict Sperner property
  • isUpperSemilattice(Poset) -- see isUpperSemilattice -- determines if a poset is an upper (or join) semilattice
  • isUpperSemimodular(Poset) -- see isUpperSemimodular -- determines if a lattice is upper semimodular
  • joinExists(Poset,Thing,Thing) -- see joinExists -- determines if the join exists for two elements of a poset
  • joinIrreducibles(Poset) -- see joinIrreducibles -- determines the join irreducible elements of a poset
  • linearExtensions(Poset) -- see linearExtensions -- computes all linear extensions of a poset
  • magnitude(Poset) -- see magnitude -- computes the magnitude of a poset
  • maximalAntichains(Poset) -- see maximalAntichains -- computes all maximal antichains of a poset
  • maximalChains(Poset) -- see maximalChains -- computes all maximal chains of a poset
  • maximalElements(Poset) -- see maximalElements -- determines the maximal elements of a poset
  • meetExists(Poset,Thing,Thing) -- see meetExists -- determines if the meet exists for two elements of a poset
  • meetIrreducibles(Poset) -- see meetIrreducibles -- determines the meet irreducible elements of a poset
  • minimalElements(Poset) -- see minimalElements -- determines the minimal elements of a poset
  • moebiusFunction(Poset) -- see moebiusFunction -- computes the Moebius function at every pair of elements of a poset
  • orderComplex(Poset) -- see orderComplex -- produces the order complex of a poset
  • orderIdeal(Poset,List) -- see orderIdeal -- computes the elements below given elements in a poset
  • outputTexPoset(Poset,String) -- see outputTexPoset -- writes a LaTeX file with a TikZ-representation of a poset
  • poincare(Poset) -- see poincarePolynomial -- computes the Poincaré polynomial of a ranked poset with a unique minimal element
  • poincarePolynomial(Poset) -- see poincarePolynomial -- computes the Poincaré polynomial of a ranked poset with a unique minimal element
  • Poset _ List -- returns elements of the ground set
  • Poset _ ZZ -- returns an element of the ground set
  • Poset _* -- returns the ground set of a poset
  • vertices(Poset) -- see Poset _* -- returns the ground set of a poset
  • posetJoin(Poset,Thing,Thing) -- see posetJoin -- determines the join for two elements of a poset
  • posetMeet(Poset,Thing,Thing) -- see posetMeet -- determines the meet for two elements of a poset
  • pPartitionRing(Poset) -- see pPartitionRing -- produces the p-partition ring of a poset
  • principalFilter(Poset,Thing) -- see principalFilter -- computes the elements above a given element in a poset
  • principalOrderIdeal(Poset,Thing) -- see principalOrderIdeal -- computes the elements below a given element in a poset
  • Poset * Poset -- see product(Poset,Poset) -- computes the product of two posets
  • rankFunction(Poset) -- see rankFunction -- computes the rank function of a ranked poset
  • rankGeneratingFunction(Poset) -- see rankGeneratingFunction -- computes the rank generating function of a ranked poset
  • rank(Poset) -- see rankPoset -- generates a list of lists representing the ranks of a ranked poset
  • rankPoset(Poset) -- see rankPoset -- generates a list of lists representing the ranks of a ranked poset
  • tex(Poset) -- see texPoset -- generates a string containing a TikZ-figure of a poset
  • texPoset(Poset) -- see texPoset -- generates a string containing a TikZ-figure of a poset
  • tuttePolynomial(Poset) -- see tuttePolynomial -- computes the Tutte polynomial of a poset
  • vertexSet(Poset) (missing documentation)
  • zetaPolynomial(Poset) -- see zetaPolynomial -- computes the zeta polynomial of a poset

For the programmer

The object Poset is a type, with ancestor classes HashTable < Thing.

The source of this document is in Posets.m2:1959:0.