NumericalSemigroups -- Invariants of numerical semigroups

Description

Numerical semigroups are cofinite subsets of the natural numbers that are closed under sums. We generally refer to these simply as semigroups. A semigroup S thus includes the empty sum, 0, but we input semigroups by giving generators, all nonzero. The smallest nonzero element of S is the multiplicity. The Apery set (really sequence) of a semigroup S is the the list {a_1..a_m-1} where a_i is the smallest element in S such that a_i = i mod m. The conductor is 1 plus the largest element not in S. We generally specify a semigroup by giving a list of positive integers L with gcd = 1, representing the semigroup of all sums of elements of L.

Combinatorics of Semigroups

apery -- Compute the apery set, multiplicity and conductor
gaps -- The gap sequence of a semigroup
sums -- sum of two sequences
isGapSequence -- test whether a list of integers can be the list of gaps of a semigroup
isSymmetric -- test whether the semigroup generated by L is symmetric
weight -- weight of a semigroup
effectiveWeight -- Effective weight of a semigroup (Pflueger)

Working with the Kunz cone

aperyConeEquations -- Inequalities defining the Kunz cones
muConeEquations -- Inequalities defining the Kunz cones
mu -- Compute the point representing a semigroup in the Kunz cone
semigroupFromMu -- Inverse of the function mu
facetRays -- computes the rays spanning the face in which a semigroup lies
coneRays -- All the rays of the (homogeneous) Kunz cone
allSemigroups -- Compute the Hilbert basis and module generators of a cone of semigroups
semigroupsFromMatrix -- applies semigroupFromMu to the columns of a matrix
randomSemigroup -- Random semingroup on a given face of the Kunz cone
findSemigroups -- Find all semigroups with a given number of gaps, multiplicity and/or conductor
buchweitzCriterion -- Does L satisfies the Buchweitz criterion?
buchweitz -- An example of a semigroup that is not a Weierstrass semigroup
buchweitzSemigroups -- Finds semigroups that are not Weierstrass semigroups by the Buchweitz test

Properties of semigroup rings

burchIndex -- Compute the burchIndex of the Burch ring of a semigroup
semigroupRing -- forms the semigroup ring over "BaseField"
semigroupIdeal -- The ideal defining the semigroup ring
socle -- elements of the semigroup that are in the socle mod the multiplicity
type -- type of the local semigroup ring
genus -- genus of an object
kunzRing -- artinian reduction of a semigroup ring
kunzPoset (missing documentation)
syzFormat (missing documentation)
reduceByList (missing documentation)

Arf Semigroups

isArf -- test whether a numerical semigroup is Arf
arfIndex -- Computes the Arf index of a numerical semigroup
arfClosure -- Compute the Arf closure of a numerical semigroup
fractionalIdeal -- turn a fractional ideal into a proper ideal
infinitelyNearSemigroups -- The sequence of blowup semigroups of a numerical semigroup
infinitelyNearModules -- The sequence of blowups of a semigroup ring as fractional ideals
isMinimalMultiplicity (missing documentation)

Authors

David Eisenbud <de@berkeley.edu>
Frank-Olaf Schreyer <schreyer@math.uni-sb.de>
Theodore Wittmer Lysek <tlysek44@gmail.com>

Version

This documentation describes version 1.0 of NumericalSemigroups, released June 8, 2026.

Citation

If you have used this package in your research, please cite it as follows:

@misc{NumericalSemigroupsSource,
  title = {{NumericalSemigroups: A \emph{Macaulay2} package. Version~1.0}},
  author = {David Eisenbud and Frank-Olaf Schreyer and Theodore Wittmer Lysek},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- allSemigroups -- Compute the Hilbert basis and module generators of a cone of semigroups
- apery -- Compute the apery set, multiplicity and conductor
- aperyConeEquations -- Inequalities defining the Kunz cones
- muConeEquations -- see aperyConeEquations -- Inequalities defining the Kunz cones
- aperyFacetRays (missing documentation)
- aperySemigroupRing -- computes the semigroup ring using both the multiplicity and the full Apery set
- aperySet -- Compute the apery set of a numerical semigroup
- arfClosure -- Compute the Arf closure of a numerical semigroup
- arfIndex -- Computes the Arf index of a numerical semigroup
- buchweitz -- An example of a semigroup that is not a Weierstrass semigroup
- buchweitzCriterion -- Does L satisfies the Buchweitz criterion?
- buchweitzSemigroups -- Finds semigroups that are not Weierstrass semigroups by the Buchweitz test
- burchIndex -- Compute the burchIndex of the Burch ring of a semigroup
- coneRays -- All the rays of the (homogeneous) Kunz cone
- def1 -- degrees of a basis of T^1
- effectiveWeight -- see ewt -- Effective weight of a semigroup (Pflueger)
- ewt -- Effective weight of a semigroup (Pflueger)
- facetRays -- computes the rays spanning the face in which a semigroup lies
- findSemigroups -- Find all semigroups with a given number of gaps, multiplicity and/or conductor
- fractionalIdeal -- turn a fractional ideal into a proper ideal
- gaps -- The gap sequence of a semigroup
- infinitelyNearModules -- The sequence of blowups of a semigroup ring as fractional ideals
- infinitelyNearSemigroups -- The sequence of blowup semigroups of a numerical semigroup
- isArf -- test whether a numerical semigroup is Arf
- isGapSequence -- test whether a list of integers can be the list of gaps of a semigroup
- isKnownExample -- Is L a known Weierstrass semigroup?
- isMinimalMultiplicity (missing documentation)
- isSymmetric -- test whether the semigroup generated by L is symmetric
- kunzMatrix -- determine the set of facet equations satisfied by a semigroup
- kunzPoset (missing documentation)
- kunzRing -- artinian reduction of a semigroup ring
- mu -- Compute the point representing a semigroup in the Kunz cone
- randomSemigroup -- Random semingroup on a given face of the Kunz cone
- semigroup -- Compute the semigroup generated by a list of positive integers
- semigroupFromMu -- Inverse of the function mu
- semigroupIdeal -- The ideal defining the semigroup ring
- semigroupRing -- forms the semigroup ring over "BaseField"
- semigroupsFromMatrix -- applies semigroupFromMu to the columns of a matrix
- socle -- elements of the semigroup that are in the socle mod the multiplicity
- sums -- sum of two sequences
- syzFormat (missing documentation)
- type -- type of the local semigroup ring
- weight -- weight of a semigroup
Methods
- allSemigroups(List) -- see allSemigroups -- Compute the Hilbert basis and module generators of a cone of semigroups
- allSemigroups(ZZ) -- see allSemigroups -- Compute the Hilbert basis and module generators of a cone of semigroups
- apery(List) -- see apery -- Compute the apery set, multiplicity and conductor
- aperyConeEquations(List) -- see aperyConeEquations -- Inequalities defining the Kunz cones
- aperyConeEquations(ZZ) -- see aperyConeEquations -- Inequalities defining the Kunz cones
- muConeEquations(List) -- see aperyConeEquations -- Inequalities defining the Kunz cones
- muConeEquations(ZZ) -- see aperyConeEquations -- Inequalities defining the Kunz cones
- aperyFacetRays(List) (missing documentation)
- aperySemigroupRing(List) -- see aperySemigroupRing -- computes the semigroup ring using both the multiplicity and the full Apery set
- aperySet(HashTable) -- see aperySet -- Compute the apery set of a numerical semigroup
- aperySet(List) -- see aperySet -- Compute the apery set of a numerical semigroup
- arfClosure(List) -- see arfClosure -- Compute the Arf closure of a numerical semigroup
- arfIndex(List) -- see arfIndex -- Computes the Arf index of a numerical semigroup
- buchweitz(ZZ) -- see buchweitz -- An example of a semigroup that is not a Weierstrass semigroup
- buchweitzCriterion(List) -- see buchweitzCriterion -- Does L satisfies the Buchweitz criterion?
- buchweitzCriterion(ZZ,List) -- see buchweitzCriterion -- Does L satisfies the Buchweitz criterion?
- buchweitzSemigroups(ZZ) -- see buchweitzSemigroups -- Finds semigroups that are not Weierstrass semigroups by the Buchweitz test
- buchweitzSemigroups(ZZ,ZZ) -- see buchweitzSemigroups -- Finds semigroups that are not Weierstrass semigroups by the Buchweitz test
- buchweitzSemigroups(ZZ,ZZ,ZZ) -- see buchweitzSemigroups -- Finds semigroups that are not Weierstrass semigroups by the Buchweitz test
- burchIndex(List) -- see burchIndex -- Compute the burchIndex of the Burch ring of a semigroup
- conductor(List) -- conductor of a semigroup
- coneRays(ZZ) -- see coneRays -- All the rays of the (homogeneous) Kunz cone
- def1(List) -- see def1 -- degrees of a basis of T^1
- effectiveWeight(List) -- see ewt -- Effective weight of a semigroup (Pflueger)
- ewt(List) -- see ewt -- Effective weight of a semigroup (Pflueger)
- facetRays(List) -- see facetRays -- computes the rays spanning the face in which a semigroup lies
- findSemigroups(ZZ) -- see findSemigroups -- Find all semigroups with a given number of gaps, multiplicity and/or conductor
- findSemigroups(ZZ,ZZ) -- see findSemigroups -- Find all semigroups with a given number of gaps, multiplicity and/or conductor
- findSemigroups(ZZ,ZZ,ZZ) -- see findSemigroups -- Find all semigroups with a given number of gaps, multiplicity and/or conductor
- fractionalIdeal(List,List) -- see fractionalIdeal -- turn a fractional ideal into a proper ideal
- fractionalIdeal(Ring,List) (missing documentation)
- gaps(List) -- see gaps -- The gap sequence of a semigroup
- genus(List) -- Compute the number of gaps (genus) of a semigroup
- infinitelyNearModules(Ring) -- see infinitelyNearModules -- The sequence of blowups of a semigroup ring as fractional ideals
- infinitelyNearSemigroups(List) -- see infinitelyNearSemigroups -- The sequence of blowup semigroups of a numerical semigroup
- isArf(List) -- see isArf -- test whether a numerical semigroup is Arf
- isGapSequence(List) -- see isGapSequence -- test whether a list of integers can be the list of gaps of a semigroup
- isKnownExample(List) -- see isKnownExample -- Is L a known Weierstrass semigroup?
- isSymmetric(List) -- see isSymmetric -- test whether the semigroup generated by L is symmetric
- kunzMatrix(HashTable) -- see kunzMatrix -- determine the set of facet equations satisfied by a semigroup
- kunzMatrix(List) -- see kunzMatrix -- determine the set of facet equations satisfied by a semigroup
- kunzPoset(List) (missing documentation)
- kunzRing(List) -- see kunzRing -- artinian reduction of a semigroup ring
- mingens(List) -- Find a mininmal set of semigroup generators
- mu(HashTable) -- see mu -- Compute the point representing a semigroup in the Kunz cone
- mu(List) -- see mu -- Compute the point representing a semigroup in the Kunz cone
- randomSemigroup(List,ZZ) -- see randomSemigroup -- Random semingroup on a given face of the Kunz cone
- randomSemigroup(ZZ,ZZ) -- see randomSemigroup -- Random semingroup on a given face of the Kunz cone
- semigroup(List) -- see semigroup -- Compute the semigroup generated by a list of positive integers
- semigroupFromMu(List) -- see semigroupFromMu -- Inverse of the function mu
- semigroupIdeal(List) -- see semigroupIdeal -- The ideal defining the semigroup ring
- semigroupRing(List) -- see semigroupRing -- forms the semigroup ring over "BaseField"
- semigroupsFromMatrix(Matrix) -- see semigroupsFromMatrix -- applies semigroupFromMu to the columns of a matrix
- socle(List) -- see socle -- elements of the semigroup that are in the socle mod the multiplicity
- sums(List,List) -- see sums -- sum of two sequences
- sums(ZZ,List) -- see sums -- sum of two sequences
- syzFormat(List) (missing documentation)
- type(List) -- see type -- type of the local semigroup ring
- weight(List) -- see weight -- weight of a semigroup
Symbols
- muFacetRays (missing documentation)
- reduceByList (missing documentation)

For the programmer

The object NumericalSemigroups is a package, defined in NumericalSemigroups.m2.

The source of this document is in NumericalSemigroups.m2:1545:0.