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# numericalsgps-- a package for numerical semigroups

Version 1.1.10

Manuel Delgado
Email: mdelgado@fc.up.pt
Homepage: http://www.fc.up.pt/cmup/mdelgado

Pedro A. García-Sánchez
Email: pedro@ugr.es
Homepage: http://www.ugr.es/~pedro

José João Morais

### Copyright

© 2005--2015 Centro de Matemática da Universidade do Porto, Portugal and Universidad de Granada, Spain

Numericalsgps is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the file 'GPL' included in the package or see the FSF's own site.

### Acknowledgements

The first author's work was (partially) supported by the Centro de Matemática da Universidade do Porto (CMUP), financed by FCT (Portugal) through the programs POCTI (Programa Operacional "Ciência, Tecnologia, Inovação") and POSI (Programa Operacional Sociedade da Informação), with national and European Community structural funds and a sabbatical grant of FCT.

The second author was supported by the projects MTM2004-01446, MTM2007-62346 and MTM2010-15595, the Junta de Andalucía group FQM-343, and FEDER founds.

The third author acknowledges financial support of FCT and the POCTI program through a scholarship given by Centro de Matemática da Universidade do Porto.

The authors wish to thank J. I. García-García and Alfredo Sánchez-R. Navarro for many helpful discussions and for helping in the programming of preliminary versions of some functions, and also to C. O'Neill, A. Sammartano, I. Ojeda, C. M. Moreno Ávila, A. Herrera-Poyatos and K. Stokes for their contributions (see Contributions Chapter). We are also in debt with S. Gutsche, M. Horn, H. Schönemann, C. Söeger and M. Barakat for their fruitful advices concerning 4ti2Interface, SingularInterface, Singular, Normaliz, NormalizInterface and GradedModules.

The first and second authors warmly thank María Burgos for her support and help.

Concerning the mantainment:

The first author was (partially) supported by the FCT project PTDC/MAT/65481/2006 and also by the Centro de Matemática da Universidade do Porto (CMUP), funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011.

The second author was/is supported by the project MTM2014-55367-P, which is funded by Ministerio de Economía y Competitividad and the Fondo Europeo de Desarrollo Regional FEDER.

Both maintainers want to acknowledge partial support by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.

Both maintainers are also supported by the project MTM2014-55367-P, which is funded by Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional FEDER.

The maintainers want to thank the organizers of `GAPDays` in their several editions.

The authors also thank the Centro de Servicios de Informática y Redes de Comunicaciones (CSIRC), Universidad de Granada, for providing the computing time, specially Rafael Arco Arredondo for installing this package and the extra software needed in alhambra.ugr.es.

### Colophon

This work started when (in 2004) the first author visited the University of Granada in part of a sabbatical year. Since Version 0.96 (released in 2008), the package is maintained by the first two authors. Bug reports, suggestions and comments are, of course, welcome. Please use our email addresses to this effect.

If you have benefited from the use of the numerigalsgps GAP package in your research, please cite it in addition to GAP itself, following the scheme proposed in http://www.gap-system.org/Contacts/cite.html.

If you have predominantly used the functions in the Appendix, contributed by other authors, please cite in addition these authors, referring "software implementations available in the GAP package NumericalSgps".

### Contents

1 Introduction
2 Numerical Semigroups
2.1 Generating Numerical Semigroups
2.1-1 NumericalSemigroup
2.1-2 NumericalSemigroupBySubAdditiveFunction
2.1-3 NumericalSemigroupByAperyList
2.1-4 NumericalSemigroupBySmallElements
2.1-5 NumericalSemigroupByGaps
2.1-6 NumericalSemigroupByFundamentalGaps
2.1-7 NumericalSemigroupByAffineMap
2.1-8 ModularNumericalSemigroup
2.1-9 ProportionallyModularNumericalSemigroup
2.1-10 NumericalSemigroupByInterval
2.1-11 NumericalSemigroupByOpenInterval
2.2 Some basic tests
2.2-1 IsNumericalSemigroup
2.2-2 RepresentsSmallElementsOfNumericalSemigroup
2.2-3 RepresentsGapsOfNumericalSemigroup
2.2-4 IsAperyListOfNumericalSemigroup
2.2-5 IsSubsemigroupOfNumericalSemigroup
2.2-6 IsSubset
2.2-7 BelongsToNumericalSemigroup
3 Basic operations with numerical semigroups
3.1 Invariants
3.1-1 Multiplicity
3.1-2 Generators
3.1-3 EmbeddingDimension
3.1-4 SmallElements
3.1-5 FirstElementsOfNumericalSemigroup
`3.1-6 \[ \]`
`3.1-7 \{ \}`
3.1-8 NextElementOfNumericalSemigroup
3.1-9 ElementNumber_NumericalSemigroup
3.1-10 RthElementOfNumericalSemigroup
3.1-11 NumberElement_NumericalSemigroup
3.1-12 Iterator
3.1-13 AperyList
3.1-14 AperyList
3.1-15 AperyList
3.1-16 AperyListOfNumericalSemigroupAsGraph
3.1-17 KunzCoordinates
3.1-18 KunzPolytope
3.1-19 CocycleOfNumericalSemigroupWRTElement
3.1-20 FrobeniusNumber
3.1-21 Conductor
3.1-22 PseudoFrobenius
3.1-23 Type
3.1-24 Gaps
3.1-25 Weight
3.1-26 Deserts
3.1-27 IsOrdinary
3.1-28 IsAcute
3.1-29 Holes
3.1-30 LatticePathAssociatedToNumericalSemigroup
3.1-31 Genus
3.1-32 FundamentalGaps
3.1-33 SpecialGaps
3.2 Wilf's conjecture
3.2-1 WilfNumber
3.2-2 EliahouNumber
3.2-3 ProfileOfNumericalSemigroup
3.2-4 EliahouSlicesOfNumericalSemigroup
4 Presentations of Numerical Semigroups
4.1 Presentations of Numerical Semigroups
4.1-1 MinimalPresentation
4.1-2 GraphAssociatedToElementInNumericalSemigroup
4.1-3 BettiElements
4.1-4 DegreesOfPrimitiveElementsOfNumericalSemigroup
4.1-5 ShadedSetOfElementInNumericalSemigroup
4.2 Uniquely Presented Numerical Semigroups
4.2-1 IsUniquelyPresented
4.2-2 IsGeneric
5 Constructing numerical semigroups from others
5.1 Adding and removing elements of a numerical semigroup
5.1-1 RemoveMinimalGeneratorFromNumericalSemigroup
5.1-2 AddSpecialGapOfNumericalSemigroup
5.2 Intersections, and quotients and multiples by integers
5.2-1 Intersection
5.2-2 QuotientOfNumericalSemigroup
5.2-3 MultipleOfNumericalSemigroup
5.2-4 Difference
5.2-5 NumericalDuplication
5.2-6 InductiveNumericalSemigroup
5.3 Constructing the set of all numerical semigroups containing a given numerical semigroup
5.3-1 OverSemigroups
5.4 Constructing the set of numerical semigroup with given Frobenius number
5.4-1 NumericalSemigroupsWithFrobeniusNumber
5.5 Constructing the set of numerical semigroups with genus g
5.5-1 NumericalSemigroupsWithGenus
5.6 Constructing the set of numerical semigroups with a given set of pseudo-Frobenius numbers
5.6-1 ForcedIntegersForPseudoFrobenius
5.6-2 SimpleForcedIntegersForPseudoFrobenius
5.6-3 NumericalSemigroupsWithPseudoFrobeniusNumbers
5.6-4 ANumericalSemigroupWithPseudoFrobeniusNumbers
6 Irreducible numerical semigroups
6.1 Irreducible numerical semigroups
6.1-1 IsIrreducible
6.1-2 IsSymmetric
6.1-3 IsPseudoSymmetric
6.1-4 AnIrreducibleNumericalSemigroupWithFrobeniusNumber
6.1-5 IrreducibleNumericalSemigroupsWithFrobeniusNumber
6.1-6 DecomposeIntoIrreducibles
6.2 Complete intersection numerical semigroups
6.2-1 AsGluingOfNumericalSemigroups
6.2-2 IsCompleteIntersection
6.2-3 CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber
6.2-4 IsFree
6.2-5 FreeNumericalSemigroupsWithFrobeniusNumber
6.2-6 IsTelescopic
6.2-7 TelescopicNumericalSemigroupsWithFrobeniusNumber
6.2-8 IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity
6.2-9 NumericalSemigroupsPlanarSingularityWithFrobeniusNumber
6.2-10 IsAperySetGammaRectangular
6.2-11 IsAperySetBetaRectangular
6.2-12 IsAperySetAlphaRectangular
6.3 Almost-symmetric numerical semigroups
6.3-1 AlmostSymmetricNumericalSemigroupsFromIrreducible
6.3-2 AlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType
6.3-3 IsAlmostSymmetric
6.3-4 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber
6.3-5 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType
7 Ideals of numerical semigroups
7.1 Definitions and basic operations
7.1-1 IdealOfNumericalSemigroup
7.1-2 IsIdealOfNumericalSemigroup
7.1-3 MinimalGenerators
7.1-4 Generators
7.1-5 AmbientNumericalSemigroupOfIdeal
7.1-6 IsIntegral
7.1-7 SmallElements
7.1-8 Conductor
7.1-9 Minimum
7.1-10 BelongsToIdealOfNumericalSemigroup
7.1-11 ElementNumber_IdealOfNumericalSemigroup
7.1-12 NumberElement_IdealOfNumericalSemigroup
`7.1-13 \[ \]`
`7.1-14 \{ \}`
7.1-15 Iterator
7.1-16 SumIdealsOfNumericalSemigroup
7.1-17 MultipleOfIdealOfNumericalSemigroup
7.1-18 SubtractIdealsOfNumericalSemigroup
7.1-19 Difference
7.1-20 TranslationOfIdealOfNumericalSemigroup
7.1-21 Intersection
7.1-22 MaximalIdeal
7.1-23 CanonicalIdeal
7.1-24 IsCanonicalIdeal
7.1-25 TypeSequence
7.2 Blow ups and closures
7.2-1 HilbertFunctionOfIdealOfNumericalSemigroup
7.2-2 HilbertFunction
7.2-3 BlowUp
7.2-4 ReductionNumber
7.2-5 BlowUp
7.2-6 LipmanSemigroup
7.2-7 RatliffRushNumber
7.2-8 RatliffRushClosure
7.2-9 AsymptoticRatliffRushNumber
7.2-10 MultiplicitySequence
7.2-11 MicroInvariants
7.2-12 AperyList
7.2-13 AperyTable
7.2-14 StarClosureOfIdealOfNumericalSemigroup
7.3 Patterns for ideals
7.3-1 IsAdmissiblePattern
7.3-2 IsStronglyAdmissiblePattern
7.3-3 AsIdealOfNumericalSemigroup
7.3-4 BoundForConductorOfImageOfPattern
7.3-5 ApplyPatternToIdeal
7.3-6 ApplyPatternToNumericalSemigroup
7.3-7 IsAdmittedPatternByIdeal
7.3-8 IsAdmittedPatternByNumericalSemigroup
7.4 Graded associated ring of numerical semigroup
7.4-1 IsGradedAssociatedRingNumericalSemigroupCM
7.4-2 IsGradedAssociatedRingNumericalSemigroupBuchsbaum
7.4-3 TorsionOfAssociatedGradedRingNumericalSemigroup
7.4-4 BuchsbaumNumberOfAssociatedGradedRingNumericalSemigroup
7.4-5 IsMpure
7.4-6 IsPure
7.4-7 IsGradedAssociatedRingNumericalSemigroupGorenstein
7.4-8 IsGradedAssociatedRingNumericalSemigroupCI
8 Numerical semigroups with maximal embedding dimension
8.1 Numerical semigroups with maximal embedding dimension
8.1-1 IsMED
8.1-2 MEDClosure
8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup
8.2 Numerical semigroups with the Arf property and Arf closures
8.2-1 IsArf
8.2-2 ArfClosure
8.2-3 ArfCharactersOfArfNumericalSemigroup
8.2-4 ArfNumericalSemigroupsWithFrobeniusNumber
8.2-5 ArfNumericalSemigroupsWithFrobeniusNumberUpTo
8.2-6 ArfNumericalSemigroupsWithGenus
8.2-7 ArfNumericalSemigroupsWithGenusUpTo
8.2-8 ArfNumericalSemigroupsWithGenusAndFrobeniusNumber
8.3 Saturated numerical semigroups
8.3-1 IsSaturated
8.3-2 SaturatedClosure
8.3-3 SaturatedNumericalSemigroupsWithFrobeniusNumber
9 Nonunique invariants for factorizations in numerical semigroups
9.1 Factorizations in Numerical Semigroups
9.1-1 FactorizationsIntegerWRTList
9.1-2 Factorizations
9.1-3 FactorizationsElementListWRTNumericalSemigroup
9.1-4 RClassesOfSetOfFactorizations
9.1-5 LShapes
9.1-6 DenumerantOfElementInNumericalSemigroup
9.1-7 DenumerantFunction
9.2 Invariants based on lengths
9.2-1 LengthsOfFactorizationsIntegerWRTList
9.2-2 LengthsOfFactorizationsElementWRTNumericalSemigroup
9.2-3 Elasticity
9.2-4 Elasticity
9.2-5 DeltaSet
9.2-6 DeltaSet
9.2-7 DeltaSetPeriodicityBoundForNumericalSemigroup
9.2-8 DeltaSetPeriodicityStartForNumericalSemigroup
9.2-9 DeltaSetListUpToElementWRTNumericalSemigroup
9.2-10 DeltaSetUnionUpToElementWRTNumericalSemigroup
9.2-11 DeltaSet
9.2-12 MaximumDegree
9.2-13 MaximalDenumerant
9.2-14 MaximalDenumerantOfSetOfFactorizations
9.2-15 MaximalDenumerant
9.2-16 Adjustment
9.2-17 IsAdditiveNumericalSemigroup
9.2-18 IsSuperSymmetricNumericalSemigroup
9.3 Invariants based on distances
9.3-1 CatenaryDegree
9.3-2 AdjacentCatenaryDegreeOfSetOfFactorizations
9.3-3 EqualCatenaryDegreeOfSetOfFactorizations
9.3-4 MonotoneCatenaryDegreeOfSetOfFactorizations
9.3-5 CatenaryDegree
9.3-6 TameDegree
9.3-7 CatenaryDegree
9.3-8 DegreesOffEqualPrimitiveElementsOfNumericalSemigroup
9.3-9 EqualCatenaryDegreeOfNumericalSemigroup
9.3-10 DegreesOfMonotonePrimitiveElementsOfNumericalSemigroup
9.3-11 MonotoneCatenaryDegreeOfNumericalSemigroup
9.3-12 TameDegree
9.3-13 TameDegree
9.4 Primality
9.4-1 OmegaPrimality
9.4-2 OmegaPrimalityOfElementListInNumericalSemigroup
9.4-3 OmegaPrimality
9.5 Homogenization of Numerical Semigroups
9.5-1 BelongsToHomogenizationOfNumericalSemigroup
9.5-2 FactorizationsInHomogenizationOfNumericalSemigroup
9.5-3 HomogeneousBettiElementsOfNumericalSemigroup
9.5-4 HomogeneousCatenaryDegreeOfNumericalSemigroup
9.6 Divisors, posets
9.6-1 MoebiusFunctionAssociatedToNumericalSemigroup
9.6-2 MoebiusFunction
9.6-3 DivisorsOfElementInNumericalSemigroup
9.7 Feng-Rao distances and numbers
9.7-1 FengRaoDistance
9.7-2 FengRaoNumber
10 Polynomials and numerical semigroups
10.1 Generating functions or Hilbert series
10.1-1 NumericalSemigroupPolynomial
10.1-2 IsNumericalSemigroupPolynomial
10.1-3 NumericalSemigroupFromNumericalSemigroupPolynomial
10.1-4 HilbertSeriesOfNumericalSemigroup
10.1-5 GraeffePolynomial
10.1-6 IsCyclotomicPolynomial
10.1-7 IsKroneckerPolynomial
10.1-8 IsCyclotomicNumericalSemigroup
10.1-9 IsSelfReciprocalUnivariatePolynomial
10.2 Semigroup of values of algebraic curves
10.2-1 SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity
10.2-2 IsDeltaSequence
10.2-3 DeltaSequencesWithFrobeniusNumber
10.2-4 CurveAssociatedToDeltaSequence
10.2-5 SemigroupOfValuesOfPlaneCurve
10.2-6 SemigroupOfValuesOfCurve_Local
10.2-7 SemigroupOfValuesOfCurve_Global
10.2-8 GeneratorsModule_Global
10.2-9 GeneratorsKahlerDifferentials
10.2-10 IsMonomialNumericalSemigroup
11 Affine semigroups
11.1 Defining affine semigroups
11.1-1 AffineSemigroup
11.1-2 AffineSemigroupByEquations
11.1-3 AffineSemigroupByInequalities
11.1-4 Generators
11.1-5 MinimalGenerators
11.1-6 AsAffineSemigroup
11.1-7 IsAffineSemigroup
11.1-8 BelongsToAffineSemigroup
11.1-9 IsFull
11.1-10 HilbertBasisOfSystemOfHomogeneousEquations
11.1-11 HilbertBasisOfSystemOfHomogeneousInequalities
11.1-12 EquationsOfGroupGeneratedBy
11.1-13 BasisOfGroupGivenByEquations
11.2 Gluings of affine semigroups
11.2-1 GluingOfAffineSemigroups
11.3 Presentations of affine semigroups
11.3-1 GeneratorsOfKernelCongruence
11.3-2 CanonicalBasisOfKernelCongruence
11.3-3 GraverBasis
11.3-4 MinimalPresentation
11.3-5 BettiElements
11.3-6 ShadedSetOfElementInAffineSemigroup
11.3-7 IsGeneric
11.3-8 IsUniquelyPresented
11.3-9 DegreesOfPrimitiveElementsOfAffineSemigroup
11.4 Factorizations in affine semigroups
11.4-1 FactorizationsVectorWRTList
11.4-2 Factorizations
11.4-3 Elasticity
11.4-4 Elasticity
11.4-5 DeltaSet
11.4-6 CatenaryDegree
11.4-7 EqualCatenaryDegreeOfAffineSemigroup
11.4-8 HomogeneousCatenaryDegreeOfAffineSemigroup
11.4-9 MonotoneCatenaryDegreeOfAffineSemigroup
11.4-10 TameDegree
11.4-11 OmegaPrimality
11.4-12 OmegaPrimality
12 Good semigroups
12.1 Defining good semigroups
12.1-1 IsGoodSemigroup
12.1-2 NumericalSemigroupDuplication
12.1-3 AmalgamationOfNumericalSemigroups
12.1-4 CartesianProductOfNumericalSemigroups
12.1-5 GoodSemigroup
12.2 Notable elements
12.2-1 BelongsToGoodSemigroup
12.2-2 Conductor
12.2-3 SmallElements
12.2-4 RepresentsSmallElementsOfGoodSemigroup
12.2-5 GoodSemigroupBySmallElements
12.2-6 MaximalElementsOfGoodSemigroup
12.2-7 IrreducibleMaximalElementsOfGoodSemigroup
12.2-8 GoodSemigroupByMaximalElements
12.2-9 MinimalGoodGenerators
12.3 Symmetric good semigroups
12.3-1 IsSymmetric
12.4 Arf good closure
12.4-1 ArfClosure
12.5 Good ideals
12.5-1 GoodIdeal
12.5-2 GoodGeneratingSystemOfGoodIdeal
12.5-3 AmbientGoodSemigroupOfGoodIdeal
12.5-4 MinimalGoodGeneratingSystemOfGoodIdeal
12.5-5 BelongsToGoodIdeal
12.5-6 SmallElements
12.5-7 CanonicalIdealOfGoodSemigroup
13 External packages
13.1 Using external packages
13.1-1 NumSgpsUse4ti2
13.1-2 NumSgpsUse4ti2gap
13.1-3 NumSgpsUseNormalize
13.1-4 NumSgpsUseSingular
13.1-5 NumSgpsUseSingularInterface
13.1-6 NumSgpsUseSingularGradedModules
14 Dot functions
14.1 Dot functions
14.1-1 DotBinaryRelation
14.1-2 HasseDiagramOfNumericalSemigroup
14.1-3 HasseDiagramOfBettiElementsOfNumericalSemigroup
14.1-4 HasseDiagramOfAperyListOfNumericalSemigroup
14.1-5 DotTreeOfGluingsOfNumericalSemigroup
14.1-6 DotOverSemigroupsNumericalSemigroup
14.1-7 DotRosalesGraph
14.1-8 DotFactorizationGraph
14.1-9 DotEliahouGraph
14.1-10 SetDotNSEngine
14.1-11 DotSplash
A Generalities
A.1 Bézout sequences
A.1-1 BezoutSequence
A.1-2 IsBezoutSequence
A.1-3 CeilingOfRational
A.2 Periodic subadditive functions
A.2-1 RepresentsPeriodicSubAdditiveFunction
A.2-2 IsListOfIntegersNS
B "Random" functions
B.1 Random functions
B.1-1 RandomNumericalSemigroup
B.1-2 RandomListForNS
B.1-3 RandomModularNumericalSemigroup
B.1-4 RandomProportionallyModularNumericalSemigroup
B.1-5 RandomListRepresentingSubAdditiveFunction
B.1-6 NumericalSemigroupWithRandomElementsAndFrobenius
C Contributions
C.1 Functions implemented by A. Sammartano
C.2 Functions implemented by C. O'Neill
C.3 Functions implemented by K. Stokes
C.4 Functions implemented by I. Ojeda and C. J. Moreno Ávila
C.5 Functions implemented by I. Ojeda
C.6 Functions implemented by A. Sánchez-R. Navarro
C.7 Functions implemented by G. Zito
C.8 Functions implemented by A. Herrera-Poyatos
C.9 Functions implemented by Benjamin Heredia
C.10 Functions implemented by Juan Ignacio García-García
References
Index
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