Macaulay2
»
Documentation
Packages
»
Posets
::
Index
next | previous | forward | backward | up |
index
|
toc
Posets : Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
adjoinMax
-- computes the poset with a new maximum element
adjoinMax(Poset)
-- computes the poset with a new maximum element
adjoinMax(Poset,Thing)
-- computes the poset with a new maximum element
adjoinMin
-- computes the poset with a new minimum element
adjoinMin(Poset)
-- computes the poset with a new minimum element
adjoinMin(Poset,Thing)
-- computes the poset with a new minimum element
allRelations
-- computes all relations of a poset
allRelations(Poset)
-- computes all relations of a poset
allRelations(Poset,Boolean)
-- computes all relations of a poset
antichains
-- computes all antichains of a poset
antichains(Poset)
-- computes all antichains of a poset
antichains(Poset,ZZ)
-- computes all antichains of a poset
AntisymmetryStrategy
-- creates a new Poset object
areIsomorphic
-- determines if two posets are isomorphic
areIsomorphic(Poset,Poset)
-- determines if two posets are isomorphic
atoms
-- computes the list of elements covering the minimal elements of a poset
atoms(Poset)
-- computes the list of elements covering the minimal elements of a poset
augmentPoset
-- computes the poset with an adjoined minimum and maximum
augmentPoset(Poset)
-- computes the poset with an adjoined minimum and maximum
augmentPoset(Poset,Thing,Thing)
-- computes the poset with an adjoined minimum and maximum
Bias
-- generates a random poset with a given relation probability
booleanLattice
-- generates the boolean lattice on $n$ elements
booleanLattice(ZZ)
-- generates the boolean lattice on $n$ elements
boundedRegions
-- computes the number of bounded regions a hyperplane arrangement divides the space into
boundedRegions(List,Ring)
-- computes the number of bounded regions a hyperplane arrangement divides the space into
chain
-- generates the chain poset on $n$ elements
chain(ZZ)
-- generates the chain poset on $n$ elements
chains
-- computes all chains of a poset
chains(Poset)
-- computes all chains of a poset
chains(Poset,ZZ)
-- computes all chains of a poset
characteristicPolynomial
-- computes the characteristic polynomial of a ranked poset with a unique minimal element
characteristicPolynomial(...,VariableName=>...)
-- computes the characteristic polynomial of a ranked poset with a unique minimal element
characteristicPolynomial(Poset)
-- computes the characteristic polynomial of a ranked poset with a unique minimal element
closedInterval
-- computes the subposet contained between two points
closedInterval(Poset,Thing,Thing)
-- computes the subposet contained between two points
comparabilityGraph
-- produces the comparability graph of a poset
comparabilityGraph(Poset)
-- produces the comparability graph of a poset
compare
-- compares two elements in a poset
compare(Poset,Thing,Thing)
-- compares two elements in a poset
connectedComponents(Poset)
-- generates a list of connected components of a poset
coveringRelations
-- computes the minimal list of generating relations of a poset
coveringRelations(Poset)
-- computes the minimal list of generating relations of a poset
coxeterPolynomial
-- computes the Coxeter polynomial of a poset
coxeterPolynomial(...,VariableName=>...)
-- computes the Coxeter polynomial of a poset
coxeterPolynomial(Poset)
-- computes the Coxeter polynomial of a poset
degreePolynomial
-- computes the degree polynomial of a poset
degreePolynomial(Poset)
-- computes the degree polynomial of a poset
diamondProduct
-- computes the diamond product of two ranked posets
diamondProduct(Poset,Poset)
-- computes the diamond product of two ranked posets
dilworthLattice
-- computes the Dilworth lattice of a poset
dilworthLattice(Poset)
-- computes the Dilworth lattice of a poset
dilworthNumber
-- computes the Dilworth number of a poset
dilworthNumber(Poset)
-- computes the Dilworth number of a poset
displayPoset
-- generates a PDF representation of a poset and attempts to display it
displayPoset(...,Jitter=>...)
-- generates a PDF representation of a poset and attempts to display it
displayPoset(...,PDFDirectory=>...)
-- generates a PDF representation of a poset and attempts to display it
displayPoset(...,SuppressLabels=>...)
-- generates a PDF representation of a poset and attempts to display it
displayPoset(Poset)
-- generates a PDF representation of a poset and attempts to display it
distributiveLattice
-- computes the lattice of order ideals of a poset
distributiveLattice(Poset)
-- computes the lattice of order ideals of a poset
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
divisorPoset(ZZ)
-- generates the poset of divisors
dominanceLattice
-- generates the dominance lattice of partitions of $n$
dominanceLattice(ZZ)
-- generates the dominance lattice of partitions of $n$
dropElements
-- computes the induced subposet of a poset given a list of elements to remove
dropElements(Poset,Function)
-- computes the induced subposet of a poset given a list of elements to remove
dropElements(Poset,List)
-- computes the induced subposet of a poset given a list of elements to remove
dual(Poset)
-- produces the derived poset with relations reversed
Example: Constructing common posets
Example: Hibi ideals
Example: Intersection lattices
Example: LCM-lattices
facePoset
-- generates the face poset of a simplicial complex
facePoset(SimplicialComplex)
-- generates the face poset of a simplicial complex
filter
-- computes the elements above given elements in a poset
filter(Poset,List)
-- computes the elements above given elements in a poset
filtration
-- generates the filtration of a poset
filtration(Poset)
-- generates the filtration of a poset
flagChains
-- computes the maximal chains in a list of flags of a ranked poset
flagChains(Poset,List)
-- computes the maximal chains in a list of flags of a ranked poset
flagfPolynomial
-- computes the flag-f polynomial of a ranked poset
flagfPolynomial(...,VariableName=>...)
-- computes the flag-f polynomial of a ranked poset
flagfPolynomial(Poset)
-- computes the flag-f polynomial of a ranked poset
flaghPolynomial
-- computes the flag-h polynomial of a ranked poset
flaghPolynomial(...,VariableName=>...)
-- computes the flag-h polynomial of a ranked poset
flaghPolynomial(Poset)
-- computes the flag-h polynomial of a ranked poset
flagPoset
-- computes the subposet of specified ranks of a ranked poset
flagPoset(Poset,List)
-- computes the subposet of specified ranks of a ranked poset
fPolynomial
-- computes the f-polynomial of a poset
fPolynomial(...,VariableName=>...)
-- computes the f-polynomial of a poset
fPolynomial(Poset)
-- computes the f-polynomial of a poset
gapConvertPoset
-- converts between Macaulay2's Posets and GAP's Posets
gapConvertPoset(Array)
-- converts between Macaulay2's Posets and GAP's Posets
gapConvertPoset(Poset)
-- converts between Macaulay2's Posets and GAP's Posets
gapConvertPoset(String)
-- converts between Macaulay2's Posets and GAP's Posets
greeneKleitmanPartition
-- computes the Greene-Kleitman partition of a poset
greeneKleitmanPartition(...,Strategy=>...)
-- computes the Greene-Kleitman partition of a poset
greeneKleitmanPartition(Poset)
-- computes the Greene-Kleitman partition of a poset
GroundSet
-- a class for partially ordered sets (posets)
hasseDiagram
-- produces the Hasse diagram of a poset
hasseDiagram(Poset)
-- produces the Hasse diagram of a poset
height(Poset)
-- computes the height of a poset
hibiIdeal
-- produces the Hibi ideal of a poset
hibiIdeal(...,CoefficientRing=>...)
-- produces the Hibi ideal of a poset
hibiIdeal(Poset)
-- produces the Hibi ideal of a poset
hibiRing
-- produces the Hibi ring of a poset
hibiRing(...,CoefficientRing=>...)
-- produces the Hibi ring of a poset
hibiRing(...,Strategy=>...)
-- produces the Hibi ring of a poset
hibiRing(Poset)
-- produces the Hibi ring of a poset
hPolynomial
-- computes the h-polynomial of a poset
hPolynomial(...,VariableName=>...)
-- computes the h-polynomial of a poset
hPolynomial(Poset)
-- computes the h-polynomial of a poset
incomparabilityGraph
-- produces the incomparability graph of a poset
incomparabilityGraph(Poset)
-- produces the incomparability graph of a poset
indexLabeling
-- relabels a poset with the labeling based on the indices of the vertices
indexLabeling(Poset)
-- relabels a poset with the labeling based on the indices of the vertices
intersectionLattice
-- generates the intersection lattice of a hyperplane arrangement
intersectionLattice(List,Ring)
-- generates the intersection lattice of a hyperplane arrangement
isAntichain
-- determines if a given list of vertices is an antichain of a poset
isAntichain(Poset,List)
-- determines if a given list of vertices is an antichain of a poset
isAtomic
-- determines if a lattice is atomic
isAtomic(Poset)
-- determines if a lattice is atomic
isBounded
-- determines if a poset is bounded
isBounded(Poset)
-- determines if a poset is bounded
isComparabilityGraph
-- determines if a graph is the comparability graph of a poset
isComparabilityGraph(Graph)
-- determines if a graph is the comparability graph of a poset
isConnected(Poset)
-- determines if a poset is connected
isDistributive
-- determines if a lattice is distributive
isDistributive(Poset)
-- determines if a lattice is distributive
isEulerian(Poset)
-- determines if a ranked poset is Eulerian
isGeometric
-- determines if a lattice is geometric
isGeometric(Poset)
-- determines if a lattice is geometric
isGraded
-- determines if a poset is graded
isGraded(Poset)
-- determines if a poset is graded
isLattice
-- determines if a poset is a lattice
isLattice(Poset)
-- determines if a poset is a lattice
isLowerSemilattice
-- determines if a poset is a lower (or meet) semilattice
isLowerSemilattice(Poset)
-- determines if a poset is a lower (or meet) semilattice
isLowerSemimodular
-- determines if a ranked lattice is lower semimodular
isLowerSemimodular(Poset)
-- determines if a ranked lattice is lower semimodular
isModular
-- determines if a lattice is modular
isModular(Poset)
-- determines if a lattice is modular
isomorphism
-- computes an isomorphism between isomorphic posets
isomorphism(Poset,Poset)
-- computes an isomorphism between isomorphic posets
isRanked
-- determines if a poset is ranked
isRanked(Poset)
-- determines if a poset is ranked
isSperner
-- determines if a ranked poset has the Sperner property
isSperner(Poset)
-- determines if a ranked poset has the Sperner property
isStrictSperner
-- determines if a ranked poset has the strict Sperner property
isStrictSperner(Poset)
-- determines if a ranked poset has the strict Sperner property
isUpperSemilattice
-- determines if a poset is an upper (or join) semilattice
isUpperSemilattice(Poset)
-- determines if a poset is an upper (or join) semilattice
isUpperSemimodular
-- determines if a lattice is upper semimodular
isUpperSemimodular(Poset)
-- determines if a lattice is upper semimodular
Jitter
-- generates a string containing a TikZ-figure of a poset
joinExists
-- determines if the join exists for two elements of a poset
joinExists(Poset,Thing,Thing)
-- determines if the join exists for two elements of a poset
joinIrreducibles
-- determines the join irreducible elements of a poset
joinIrreducibles(Poset)
-- determines the join irreducible elements of a poset
labelPoset
-- relabels a poset with the specified labeling
labelPoset(Poset,HashTable)
-- relabels a poset with the specified labeling
lcmLattice
-- generates the lattice of lcms in an ideal
lcmLattice(...,Strategy=>...)
-- generates the lattice of lcms in an ideal
lcmLattice(Ideal)
-- generates the lattice of lcms in an ideal
linearExtensions
-- computes all linear extensions of a poset
linearExtensions(Poset)
-- computes all linear extensions of a poset
magnitude
-- computes the magnitude of a poset
magnitude(Poset)
-- computes the magnitude of a poset
maximalAntichains
-- computes all maximal antichains of a poset
maximalAntichains(Poset)
-- computes all maximal antichains of a poset
maximalChains
-- computes all maximal chains of a poset
maximalChains(Poset)
-- computes all maximal chains of a poset
maximalElements
-- determines the maximal elements of a poset
maximalElements(Poset)
-- determines the maximal elements of a poset
meetExists
-- determines if the meet exists for two elements of a poset
meetExists(Poset,Thing,Thing)
-- determines if the meet exists for two elements of a poset
meetIrreducibles
-- determines the meet irreducible elements of a poset
meetIrreducibles(Poset)
-- determines the meet irreducible elements of a poset
minimalElements
-- determines the minimal elements of a poset
minimalElements(Poset)
-- determines the minimal elements of a poset
moebiusFunction
-- computes the Moebius function at every pair of elements of a poset
moebiusFunction(Poset)
-- computes the Moebius function at every pair of elements of a poset
naturalLabeling
-- relabels a poset with a natural labeling
naturalLabeling(Poset)
-- relabels a poset with a natural labeling
naturalLabeling(Poset,ZZ)
-- relabels a poset with a natural labeling
NCPartition
-- generates the non-crossing partitions of size $n$
ncPartitions
-- generates the non-crossing partitions of size $n$
ncPartitions(ZZ)
-- generates the non-crossing partitions of size $n$
ncpLattice
-- computes the non-crossing partition lattice of set-partitions of size $n$
ncpLattice(ZZ)
-- computes the non-crossing partition lattice of set-partitions of size $n$
openInterval
-- computes the subposet contained strictly between two points
openInterval(Poset,Thing,Thing)
-- computes the subposet contained strictly between two points
orderComplex
-- produces the order complex of a poset
orderComplex(...,CoefficientRing=>...)
-- produces the order complex of a poset
orderComplex(...,VariableName=>...)
-- produces the order complex of a poset
orderComplex(Poset)
-- produces the order complex of a poset
orderIdeal
-- computes the elements below given elements in a poset
orderIdeal(Poset,List)
-- computes the elements below given elements in a poset
OriginalPoset
-- computes the lattice of order ideals of a poset
outputTexPoset
-- writes a LaTeX file with a TikZ-representation of a poset
outputTexPoset(...,Jitter=>...)
-- writes a LaTeX file with a TikZ-representation of a poset
outputTexPoset(...,SuppressLabels=>...)
-- writes a LaTeX file with a TikZ-representation of a poset
outputTexPoset(Poset,String)
-- writes a LaTeX file with a TikZ-representation of a poset
partitionLattice
-- computes the lattice of set-partitions of size $n$
partitionLattice(ZZ)
-- computes the lattice of set-partitions of size $n$
PDFDirectory
-- generates a PDF representation of a poset and attempts to display it
plueckerPoset
-- computes a poset associated to the Plücker relations
plueckerPoset(ZZ)
-- computes a poset associated to the Plücker relations
poincare(Poset)
-- computes the Poincaré polynomial of a ranked poset with a unique minimal element
poincarePolynomial
-- computes the Poincaré polynomial of a ranked poset with a unique minimal element
poincarePolynomial(...,VariableName=>...)
-- computes the Poincaré polynomial of a ranked poset with a unique minimal element
poincarePolynomial(Poset)
-- computes the Poincaré polynomial of a ranked poset with a unique minimal element
Poset
-- a class for partially ordered sets (posets)
poset
-- creates a new Poset object
Poset * Poset
-- computes the product of two posets
Poset + Poset
-- computes the union of two posets
Poset - List
-- computes the induced subposet of a poset given a list of elements to remove
Poset == Poset
-- determines if two posets are isomorphic
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
poset(...,AntisymmetryStrategy=>...)
-- creates a new Poset object
poset(List)
-- creates a new Poset object
poset(List,Function)
-- creates a new Poset object
poset(List,List)
-- creates a new Poset object
poset(List,List,Matrix)
-- creates a new Poset object
posetJoin
-- determines the join for two elements of a poset
posetJoin(Poset,Thing,Thing)
-- determines the join for two elements of a poset
posetMeet
-- determines the meet for two elements of a poset
posetMeet(Poset,Thing,Thing)
-- determines the meet for two elements of a poset
Posets
-- a package for working with partially ordered sets
pPartitionRing
-- produces the p-partition ring of a poset
pPartitionRing(...,CoefficientRing=>...)
-- produces the p-partition ring of a poset
pPartitionRing(...,Strategy=>...)
-- produces the p-partition ring of a poset
pPartitionRing(Poset)
-- produces the p-partition ring of a poset
Precompute
-- a package-wide configuration that toggles precomputation
principalFilter
-- computes the elements above a given element in a poset
principalFilter(Poset,Thing)
-- computes the elements above a given element in a poset
principalOrderIdeal
-- computes the elements below a given element in a poset
principalOrderIdeal(Poset,Thing)
-- computes the elements below a given element in a poset
product(Poset,Poset)
-- computes the product of two posets
projectivizeArrangement
-- computes the intersection poset of a projectivized hyperplane arrangement
projectivizeArrangement(List,Ring)
-- computes the intersection poset of a projectivized hyperplane arrangement
randomPoset
-- generates a random poset with a given relation probability
randomPoset(...,Bias=>...)
-- generates a random poset with a given relation probability
randomPoset(List)
-- generates a random poset with a given relation probability
randomPoset(ZZ)
-- generates a random poset with a given relation probability
rank(Poset)
-- generates a list of lists representing the ranks of a ranked poset
rankFunction
-- computes the rank function of a ranked poset
rankFunction(Poset)
-- computes the rank function of a ranked poset
rankGeneratingFunction
-- computes the rank generating function of a ranked poset
rankGeneratingFunction(...,VariableName=>...)
-- computes the rank generating function of a ranked poset
rankGeneratingFunction(Poset)
-- computes the rank generating function of a ranked poset
rankPoset
-- generates a list of lists representing the ranks of a ranked poset
rankPoset(Poset)
-- 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
realRegions(List,Ring)
-- computes the number of regions a hyperplane arrangement divides the space into
RelationMatrix
-- a class for partially ordered sets (posets)
Relations
-- a class for partially ordered sets (posets)
removeIsomorphicPosets
-- returns a sub-list of non-isomorphic posets
removeIsomorphicPosets(List)
-- returns a sub-list of non-isomorphic posets
resolutionPoset
-- generates a poset from a resolution
resolutionPoset(ChainComplex)
-- generates a poset from a resolution
resolutionPoset(Ideal)
-- generates a poset from a resolution
resolutionPoset(MonomialIdeal)
-- generates a poset from a resolution
setPartition
-- computes the list of set-partitions of size $n$
setPartition(List)
-- computes the list of set-partitions of size $n$
setPartition(ZZ)
-- computes the list of set-partitions of size $n$
setPrecompute
-- sets the Precompute configuration
setPrecompute(Boolean)
-- sets the Precompute configuration
setSuppressLabels
-- sets the SuppressLabels configuration
setSuppressLabels(Boolean)
-- sets the SuppressLabels configuration
standardMonomialPoset
-- generates the poset of divisibility in the monomial basis of an ideal
standardMonomialPoset(MonomialIdeal)
-- generates the poset of divisibility in the monomial basis of an ideal
standardMonomialPoset(MonomialIdeal,ZZ,ZZ)
-- 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
subposet(Poset,List)
-- computes the induced subposet of a poset given a list of elements
SuppressLabels
-- generates a string containing a TikZ-figure of a poset
tex(Poset)
-- generates a string containing a TikZ-figure of a poset
texPoset
-- generates a string containing a TikZ-figure of a poset
texPoset(...,Jitter=>...)
-- generates a string containing a TikZ-figure of a poset
texPoset(...,SuppressLabels=>...)
-- generates a string containing a TikZ-figure of a poset
texPoset(Poset)
-- generates a string containing a TikZ-figure of a poset
transitiveClosure
-- computes the transitive closure of a set of relations
transitiveClosure(List,List)
-- computes the transitive closure of a set of relations
transitiveOrientation
-- generates a poset whose comparability graph is the given graph
transitiveOrientation(...,Random=>...)
-- generates a poset whose comparability graph is the given graph
transitiveOrientation(...,Strategy=>...)
-- generates a poset whose comparability graph is the given graph
transitiveOrientation(Graph)
-- generates a poset whose comparability graph is the given graph
tuttePolynomial
-- computes the Tutte polynomial of a poset
tuttePolynomial(Poset)
-- computes the Tutte polynomial of a poset
union
-- computes the union of two posets
union(Poset,Poset)
-- computes the union of two posets
vertices(Poset)
-- returns the ground set of a poset
youngSubposet
-- generates a subposet of Young's lattice
youngSubposet(List)
-- generates a subposet of Young's lattice
youngSubposet(List,List)
-- generates a subposet of Young's lattice
youngSubposet(ZZ)
-- generates a subposet of Young's lattice
zetaPolynomial
-- computes the zeta polynomial of a poset
zetaPolynomial(...,VariableName=>...)
-- computes the zeta polynomial of a poset
zetaPolynomial(Poset)
-- computes the zeta polynomial of a poset