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Posets :: Posets

Posets -- a package for working with partially ordered sets

Description

This package defines Poset as a new data type and provides routines which use or produce posets. A poset (partially ordered set) is a set together with a binary relation satisfying reflexivity, antisymmetry, and transitivity.

Contributors:

The following people have generously contributed code to the package: Kristine Fisher, Andrew Hoefel, Manoj Kummini, Stephen Sturgeon, and Josephine Yu.

Other acknowledgements:

A few methods in this package have been ported from John Stembridge’s Maple package implementing posets, which is available at http://www.math.lsa.umich.edu/~jrs/maple.html#posets. Such methods are noted both in the source code and in the documentation.

See also

Authors

Certification a gold star

Version 1.1.2 of this package was accepted for publication in volume 7 of the journal The Journal of Software for Algebra and Geometry on 5 June 2015, in the article Partially ordered sets in Macaulay2. That version can be obtained from the journal or from the Macaulay2 source code repository, http://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/Posets.m2, commit number 3a8d880a524f36a9668750375bb6079a7b00ea0f.

Version

This documentation describes version 1.1.2 of Posets.

Source code

The source code from which this documentation is derived is in the file Posets.m2.

Exports

  • Types
    • NCPartition, see ncPartitions -- generates the non-crossing partitions of size $n$
    • Poset -- a class for partially ordered sets (posets)
  • Functions and commands
    • adjoinMax -- computes the poset with a new maximum element
    • adjoinMin -- computes the poset with a new minimum element
    • allRelations -- computes all relations of a poset
    • antichains -- computes all antichains of a poset
    • areIsomorphic -- determines if two posets are isomorphic
    • atoms -- computes the list of elements covering the minimal elements of a poset
    • augmentPoset -- computes the poset with an adjoined minimum and maximum
    • booleanLattice -- generates the boolean lattice on $n$ elements
    • boundedRegions -- computes the number of bounded regions a hyperplane arrangement divides the space into
    • chain -- generates the chain poset on $n$ elements
    • chains -- computes all chains of a poset
    • characteristicPolynomial -- computes the characteristic polynomial of a ranked poset with a unique minimal element
    • closedInterval -- computes the subposet contained between two points
    • comparabilityGraph -- produces the comparability graph of a poset
    • compare -- compares two elements in a poset
    • coveringRelations -- computes the minimal list of generating relations of a poset
    • diamondProduct -- computes the diamond product of two ranked posets
    • dilworthLattice -- computes the Dilworth lattice of a poset
    • dilworthNumber -- computes the Dilworth number of a poset
    • displayPoset -- generates a PDF representation of a poset and attempts to display it
    • distributiveLattice -- computes the lattice of order ideals of a poset
    • divisorPoset -- generates the poset of divisors
    • dominanceLattice -- generates the dominance lattice of partitions of $n$
    • dropElements -- computes the induced subposet of a poset given a list of elements to remove
    • facePoset -- generates the face poset of a simplicial complex
    • filter -- computes the elements above given elements in a poset
    • filtration -- generates the filtration of a poset
    • flagChains -- computes the maximal chains in a list of flags of a ranked poset
    • flagfPolynomial -- computes the flag-f polynomial of a ranked poset
    • flaghPolynomial -- computes the flag-h polynomial of a ranked poset
    • flagPoset -- computes the subposet of specified ranks of a ranked poset
    • fPolynomial -- computes the f-polynomial of a poset
    • gapConvertPoset -- converts between Macaulay2's Posets and GAP's Posets
    • greeneKleitmanPartition -- computes the Greene-Kleitman partition of a poset
    • hasseDiagram -- produces the Hasse diagram of a poset
    • hibiIdeal -- produces the Hibi ideal of a poset
    • hibiRing -- produces the Hibi ring of a poset
    • hPolynomial -- computes the h-polynomial of a poset
    • incomparabilityGraph -- produces the incomparability graph of a poset
    • indexLabeling -- relabels a poset with the labeling based on the indices of the vertices
    • intersectionLattice -- generates the intersection lattice of a hyperplane arrangement
    • isAntichain -- determines if a given list of vertices is an antichain of a poset
    • isAtomic -- determines if a lattice is atomic
    • isBounded -- determines if a poset is bounded
    • isComparabilityGraph -- determines if a graph is the comparability graph of a poset
    • isDistributive -- determines if a lattice is distributive
    • isGeometric -- determines if a lattice is geometric
    • isGraded -- determines if a poset is graded
    • isLattice -- determines if a poset is a lattice
    • isLowerSemilattice -- determines if a poset is a lower (or meet) semilattice
    • isLowerSemimodular -- determines if a ranked lattice is lower semimodular
    • isModular -- determines if a lattice is modular
    • isomorphism -- computes an isomorphism between isomorphic posets
    • isRanked -- determines if a poset is ranked
    • isSperner -- determines if a ranked poset has the Sperner property
    • isStrictSperner -- determines if a ranked poset has the strict Sperner property
    • isUpperSemilattice -- determines if a poset is an upper (or join) semilattice
    • isUpperSemimodular -- determines if a lattice is upper semimoudlar
    • joinExists -- determines if the join exists for two elements of a poset
    • joinIrreducibles -- determines the join irreducible elements of a poset
    • labelPoset -- relabels a poset with the specified labeling
    • lcmLattice -- generates the lattice of lcms in an ideal
    • linearExtensions -- computes all linear extensions of a poset
    • maximalAntichains -- computes all maximal antichains of a poset
    • maximalChains -- computes all maximal chains of a poset
    • maximalElements -- determines the maximal elements of a poset
    • meetExists -- determines if the meet exists for two elements of a poset
    • meetIrreducibles -- determines the meet irreducible elements of a poset
    • minimalElements -- determines the minimal elements of a poset
    • moebiusFunction -- computes the Moebius function at every pair of elements of a poset
    • naturalLabeling -- relabels a poset with a natural labeling
    • ncPartitions -- generates the non-crossing partitions of size $n$
    • ncpLattice -- computes the non-crossing partition lattice of set-partitions of size $n$
    • openInterval -- computes the subposet contained strictly between two points
    • orderComplex -- produces the order complex of a poset
    • orderIdeal -- computes the elements below given elements in a poset
    • outputTexPoset -- writes a LaTeX file with a TikZ-representation of a poset
    • partitionLattice -- computes the lattice of set-partitions of size $n$
    • plueckerPoset -- computes a poset associated to the Pluecker relations
    • poincarePolynomial -- computes the Poincare polynomial of a ranked poset with a unique minimal element
    • poset -- creates a new Poset object
    • posetJoin -- determines the join for two elements of a poset
    • posetMeet -- determines the meet for two elements of a poset
    • pPartitionRing -- produces the p-partition ring of a poset
    • principalFilter -- computes the elements above a given element in a poset
    • principalOrderIdeal -- computes the elements below a given element in a poset
    • projectivizeArrangement -- computes the intersection poset of a projectivized hyperplane arrangement
    • randomPoset -- generates a random poset with a given relation probability
    • rankFunction -- computes the rank function of a ranked poset
    • rankGeneratingFunction -- computes the rank generating function of a ranked poset
    • rankPoset -- generates a list of lists representing the ranks of a ranked poset
    • realRegions -- computes the number of regions a hyperplane arrangement divides the space into
    • removeIsomorphicPosets -- returns a sub-list of non-isomorphic posets
    • resolutionPoset -- generates a poset from a resolution
    • setPartition -- computes the list of set-partitions of size $n$
    • setPDFViewer -- sets the default PDFViewer option
    • setPrecompute -- sets the Precompute configuration
    • setSuppressLabels -- sets the SuppressLabels configuration
    • standardMonomialPoset -- generates the poset of divisibility in the monomial basis of an ideal
    • subposet -- computes the induced subposet of a poset given a list of elements
    • texPoset -- generates a string containing a TikZ-figure of a poset
    • transitiveClosure -- computes the transitive closure of a set of relations
    • transitiveOrientation -- generates a poset whose comparability graph is the given graph
    • tuttePolynomial -- computes the Tutte polynomial of a poset
    • union -- computes the union of two posets
    • youngSubposet -- generates a subposet of Young's lattice
    • zetaPolynomial -- computes the zeta polynomial of a poset
  • Methods
    • 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
    • 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
    • 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
    • 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
    • closedInterval(Poset,Thing,Thing), see closedInterval -- computes the subposet contained between two points
    • 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
    • diamondProduct(Poset,Poset), see diamondProduct -- computes the diamond product of two ranked posets
    • dilworthLattice(Poset), see dilworthLattice -- computes the Dilworth lattice 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
    • distributiveLattice(Poset), see distributiveLattice -- computes the lattice of order ideals of a poset
    • 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
    • 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
    • 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
    • flagPoset(Poset,List), see flagPoset -- computes the subposet of specified ranks 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
    • indexLabeling(Poset), see indexLabeling -- relabels a poset with the labeling based on the indices of the vertices
    • 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), see isomorphism -- 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 semimoudlar
    • 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
    • labelPoset(Poset,HashTable), see labelPoset -- relabels a poset with the specified labeling
    • linearExtensions(Poset), see linearExtensions -- computes all linear extensions 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
    • naturalLabeling(Poset), see naturalLabeling -- relabels a poset with a natural labeling
    • naturalLabeling(Poset,ZZ), see naturalLabeling -- relabels a poset with a natural labeling
    • openInterval(Poset,Thing,Thing), see openInterval -- computes the subposet contained strictly between two points
    • 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 Poincare polynomial of a ranked poset with a unique minimal element
    • poincarePolynomial(Poset), see poincarePolynomial -- computes the Poincare 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
    • 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
    • subposet(Poset,List), see subposet -- computes the induced subposet of a poset given a list of elements
    • 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
    • Poset + Poset, see union -- computes the union of two posets
    • union(Poset,Poset), see union -- computes the union of two posets
    • vertexSet(Poset) (missing documentation)
    • zetaPolynomial(Poset), see zetaPolynomial -- computes the zeta polynomial of a poset
  • Symbols
    • PDFDirectory, see displayPoset -- generates a PDF representation of a poset and attempts to display it
    • PDFViewer, see displayPoset -- generates a PDF representation of a poset and attempts to display it
    • OriginalPoset, see distributiveLattice -- computes the lattice of order ideals of a poset
    • GroundSet, see Poset -- a class for partially ordered sets (posets)
    • RelationMatrix, see Poset -- a class for partially ordered sets (posets)
    • Relations, see Poset -- a class for partially ordered sets (posets)
    • AntisymmetryStrategy, see poset -- creates a new Poset object
    • Precompute -- a package-wide configuration that toggles precomputation
    • Bias, see randomPoset -- generates a random poset with a given relation probability
    • Jitter, see texPoset -- generates a string containing a TikZ-figure of a poset
    • SuppressLabels, see texPoset -- generates a string containing a TikZ-figure of a poset