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 relations "generating" all relations

i3 : P = poset(G, R)                 -- the poset with its relations matrix computed

o3 = Poset{cache => CacheTable{}                        }
           GroundSet => {1, 2, 3, 4}
           RelationMatrix => | 1 1 1 1 |
                             | 0 1 0 1 |
                             | 0 0 1 1 |
                             | 0 0 0 1 |
           Relations => {{1, 2}, {1, 3}, {2, 4}, {3, 4}}

o3 : Poset

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
lcmLattice(MonomialIdeal), 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 Pluecker 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(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,List,Poset,List), see areIsomorphic -- determines if two posets are isomorphic
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
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), see connectedComponents -- generates a list of connected components of a poset
coveringRelations(Poset), see coveringRelations -- computes the minimal list of generating relations 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 posett
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), see isConnected -- determines if a poset is connected
isDistributive(Poset), see isDistributive -- determines if a lattice is distributive
isEulerian(Poset), see isEulerian -- 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,List,Poset,List), see isomorphism -- computes an isomorphism between isomorphic posets
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
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
orderComplex(Poset), see orderComplex -- produces the order complex of a poset
orderIdeal(Poset,List), see orderIdeal -- computes the elements above given elements in a poset
outputTexPoset(Poset,String), see outputTexPoset -- writes a LaTeX file with a TikZ-representation of a poset
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
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
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
Poset + Poset, see union -- computes the union of two posets
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.

Poset -- a class for partially ordered sets (posets)

Description

See also

Functions and methods returning an object of class Poset :

Methods that use an object of class Poset :

For the programmer