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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

- poset -- creates a new Poset object

## 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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Posets.m2:1959:0.