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# numericalsgps-- a package for numerical semigroups

Version 1.0.1

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.

### 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, 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 and A. Sammartano 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 projects MTM2007-62346 and MTM2010-15595.

Both authors 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.

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

### 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 ModularNumericalSemigroup
2.1-3 ProportionallyModularNumericalSemigroup
2.1-4 NumericalSemigroupByGenerators
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 BelongsToNumericalSemigroup
3 Basic operations with numerical semigroups
3.1 The definitions
3.1-1 MultiplicityOfNumericalSemigroup
3.1-2 GeneratorsOfNumericalSemigroup
3.1-3 EmbeddingDimensionOfNumericalSemigroup
3.1-4 SmallElementsOfNumericalSemigroup
3.1-5 FirstElementsOfNumericalSemigroup
3.1-6 AperyListOfNumericalSemigroupWRTElement
3.1-7 AperyListOfNumericalSemigroup
3.1-8 AperyListOfNumericalSemigroupWRTInteger
3.1-9 AperyListOfNumericalSemigroupAsGraph
3.1-10 KunzCoordinatesOfNumericalSemigroup
3.1-11 KunzPolytope
3.2 Frobenius Number
3.2-1 FrobeniusNumberOfNumericalSemigroup
3.2-2 FrobeniusNumber
3.2-3 ConductorOfNumericalSemigroup
3.2-4 PseudoFrobeniusOfNumericalSemigroup
3.2-5 TypeOfNumericalSemigroup
3.3 Gaps
3.3-1 GapsOfNumericalSemigroup
3.3-2 GenusOfNumericalSemigroup
3.3-3 FundamentalGapsOfNumericalSemigroup
3.3-4 SpecialGapsOfNumericalSemigroup
4 Presentations of Numerical Semigroups
4.1 Presentations of Numerical Semigroups
4.1-1 MinimalPresentationOfNumericalSemigroup
4.1-2 GraphAssociatedToElementInNumericalSemigroup
4.1-3 BettiElementsOfNumericalSemigroup
4.1-4 PrimitiveElementsOfNumericalSemigroup
4.1-5 ShadedSetOfElementInNumericalSemigroup
4.2 Uniquely Presented Numerical Semigroups
4.2-1 IsUniquelyPresentedNumericalSemigroup
4.2-2 IsGenericNumericalSemigroup
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 by integers
5.2-1 IntersectionOfNumericalSemigroups
5.2-2 QuotientOfNumericalSemigroup
5.3 Constructing the set of all numerical semigroups containing a given numerical semigroup
5.3-1 OverSemigroupsNumericalSemigroup
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, that is, numerical semigroups with exactly g gaps
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 RandomNumericalSemigroupWithPseudoFrobeniusNumbers
6 Irreducible numerical semigroups
6.1 Irreducible numerical semigroups
6.1-1 IsIrreducibleNumericalSemigroup
6.1-2 IsSymmetricNumericalSemigroup
6.1-3 IsPseudoSymmetricNumericalSemigroup
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 IsACompleteIntersectionNumericalSemigroup
6.2-3 CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber
6.2-4 IsFreeNumericalSemigroup
6.2-5 FreeNumericalSemigroupsWithFrobeniusNumber
6.2-6 IsTelescopicNumericalSemigroup
6.2-7 TelescopicNumericalSemigroupsWithFrobeniusNumber
6.2-8 IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity
6.2-9 NumericalSemigroupsAssociatedIrreduciblePlanarCurveSingularityWithFrobeniusNumber
6.3 Almost-symmetric numerical semigroups
6.3-1 AlmostSymmetricNumericalSemigroupsFromIrreducible
6.3-2 IsAlmostSymmetricNumericalSemigroup
6.3-3 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber
7 Ideals of numerical semigroups
7.1 Definitions and basic operations
7.1-1 IdealOfNumericalSemigroup
7.1-2 IsIdealOfNumericalSemigroup
7.1-3 MinimalGeneratingSystemOfIdealOfNumericalSemigroup
7.1-4 GeneratorsOfIdealOfNumericalSemigroup
7.1-5 AmbientNumericalSemigroupOfIdeal
7.1-6 SmallElementsOfIdealOfNumericalSemigroup
7.1-7 BelongsToIdealOfNumericalSemigroup
7.1-8 SumIdealsOfNumericalSemigroup
7.1-9 MultipleOfIdealOfNumericalSemigroup
7.1-10 SubtractIdealsOfNumericalSemigroup
7.1-11 DifferenceOfIdealsOfNumericalSemigroup
7.1-12 TranslationOfIdealOfNumericalSemigroup
7.1-13 IntersectionIdealsOfNumericalSemigroup
7.1-14 MaximalIdealOfNumericalSemigroup
7.1-15 CanonicalIdealOfNumericalSemigroup
7.2 Other functions for ideals
7.2-1 HilbertFunctionOfIdealOfNumericalSemigroup
7.2-2 BlowUpIdealOfNumericalSemigroup
7.2-3 ReductionNumberIdealNumericalSemigroup
7.2-4 BlowUpOfNumericalSemigroup
7.2-5 MicroInvariantsOfNumericalSemigroup
7.2-6 IsGradedAssociatedRingNumericalSemigroupCM
7.2-7 IsMonomialNumericalSemigroup
7.2-8 AperyListOfIdealOfNumericalSemigroupWRTElement
7.2-9 AperyTableOfNumericalSemigroup
7.2-10 StarClosureOfIdealOfNumericalSemigroup
8 Numerical semigroups with maximal embedding dimension
8.1 Numerical semigroups with maximal embedding dimension
8.1-1 IsMEDNumericalSemigroup
8.1-2 MEDNumericalSemigroupClosure
8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup
8.2 Numerical semigroups with the Arf property and Arf closures
8.2-1 IsArfNumericalSemigroup
8.2-2 ArfNumericalSemigroupClosure
8.2-3 MinimalArfGeneratingSystemOfArfNumericalSemigroup
8.2-4 ArfNumericalSemigroupsWithFrobeniusNumber
8.3 Saturated numerical semigroups
8.3-1 IsSaturatedNumericalSemigroup
8.3-2 SaturatedNumericalSemigroupClosure
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 FactorizationsElementWRTNumericalSemigroup
9.1-3 RClassesOfSetOfFactorizations
9.1-4 LShapesOfNumericalSemigroup
9.1-5 DenumerantOfElementInNumericalSemigroup
9.2 Invariants based on lengths
9.2-1 LengthsOfFactorizationsIntegerWRTList
9.2-2 LengthsOfFactorizationsElementWRTNumericalSemigroup
9.2-3 ElasticityOfFactorizationsElementWRTNumericalSemigroup
9.2-4 ElasticityOfNumericalSemigroup
9.2-5 DeltaSetOfSetOfIntegers
9.2-6 DeltaSetOfFactorizationsElementWRTNumericalSemigroup
9.2-7 MaximumDegreeOfElementWRTNumericalSemigroup
9.2-8 MaximalDenumerantOfElementInNumericalSemigroup
9.2-9 MaximalDenumerantOfSetOfFactorizations
9.2-10 MaximalDenumerantOfNumericalSemigroup
9.2-11 AdjustmentOfNumericalSemigroup
9.2-12 IsAdditiveNumericalSemigroup
9.2-13 IsSuperSymmetricNumericalSemigroup
9.3 Invariants based on distances
9.3-1 CatenaryDegreeOfSetOfFactorizations
9.3-2 AdjacentCatenaryDegreeOfSetOfFactorizations
9.3-3 EqualCatenaryDegreeOfSetOfFactorizations
9.3-4 MonotoneCatenaryDegreeOfSetOfFactorizations
9.3-5 CatenaryDegreeOfElementInNumericalSemigroup
9.3-6 TameDegreeOfSetOfFactorizations
9.3-7 CatenaryDegreeOfNumericalSemigroup
9.3-8 EqualPrimitiveElementsOfNumericalSemigroup
9.3-9 EqualCatenaryDegreeOfNumericalSemigroup
9.3-10 MonotonePrimitiveElementsOfNumericalSemigroup
9.3-11 MonotoneCatenaryDegreeOfNumericalSemigroup
9.3-12 TameDegreeOfNumericalSemigroup
9.3-13 TameDegreeOfElementInNumericalSemigroup
9.4 Primality
9.4-1 OmegaPrimalityOfElementInNumericalSemigroup
9.4-2 OmegaPrimalityOfNumericalSemigroup
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
10 Polynomials and numerical semigroups
10.1 Generating functions or Hilbert series
10.1-1 NumericalSemigroupPolynomial
10.1-2 HilbertSeriesOfNumericalSemigroup
10.1-3 GraeffePolynomial
10.1-4 IsCyclotomicPolynomial
10.1-5 IsKroneckerPolynomial
10.1-6 IsCyclotomicNumericalSemigroup
10.1-7 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 SemigroupOfValuesOfCurve_Local
10.2-6 SemigroupOfValuesOfCurve_Global
11 Affine semigroups
11.1 Using external packages
11.1-1 NumSgpsUse4ti2
11.1-2 NumSgpsUse4ti2gap
11.1-3 NumSgpsUseNormalize
11.1-4 NumSgpsUseSingular
11.1-5 NumSgpsUseSingularInterface
11.1-6 NumSgpsUseSingularGradedModules
11.2 Defining affine semigroups
11.2-1 AffineSemigroup
11.2-2 AsAffineSemigroup
11.2-3 IsAffineSemigroup
11.2-4 BelongsToAffineSemigroup
11.2-5 IsFullAffineSemigroup
11.2-6 HilbertBasisOfSystemOfHomogeneousEquations
11.2-7 HilbertBasisOfSystemOfHomogeneousInequalities
11.2-8 EquationsOfGroupGeneratedBy
11.2-9 BasisOfGroupGivenByEquations
11.3 Gluings of affine semigroups
11.3-1 GluingOfAffineSemigroups
11.4 Presentations of affine semigroups
11.4-1 GeneratorsOfKernelCongruence
11.4-2 MinimalPresentationOfAffineSemigroup
11.4-3 BettiElementsOfAffineSemigroup
11.4-4 ShadedSetOfElementInAffineSemigroup
11.4-5 IsGenericAffineSemigroup
11.4-6 IsUniquelyPresentedAffineSemigroup
11.4-7 PrimitiveElementsOfAffineSemigroup
11.5 Factorizations in affine semigroups
11.5-1 FactorizationsVectorWRTList
11.5-2 ElasticityOfAffineSemigroup
11.5-3 CatenaryDegreeOfAffineSemigroup
11.5-4 EqualCatenaryDegreeOfAffineSemigroup
11.5-5 HomogeneousCatenaryDegreeOfAffineSemigroup
11.5-6 MonotoneCatenaryDegreeOfAffineSemigroup
11.5-7 TameDegreeOfAffineSemigroup
11.5-8 OmegaPrimalityOfElementInAffineSemigroup
11.5-9 OmegaPrimalityOfAffineSemigroup
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
C Contributions
C.1 Functions implemented by A. Sammartano
C.1-1 IsGradedAssociatedRingNumericalSemigroupBuchsbaum
C.1-2 IsMpureNumericalSemigroup
C.1-3 IsPureNumericalSemigroup
C.1-4 IsGradedAssociatedRingNumericalSemigroupGorenstein
C.1-5 IsGradedAssociatedRingNumericalSemigroupCI
C.1-6 IsAperySetGammaRectangular
C.1-7 IsAperySetBetaRectangular
C.1-8 IsAperySetAlphaRectangular
C.1-9 TypeSequenceOfNumericalSemigroup
C.2 Functions implemented by C. O'Neill
C.2-1 OmegaPrimalityOfElementListInNumericalSemigroup
C.2-2 FactorizationsElementListWRTNumericalSemigroup
C.2-3 DeltaSetPeriodicityBoundForNumericalSemigroup
C.2-4 DeltaSetPeriodicityStartForNumericalSemigroup
C.2-5 DeltaSetListUpToElementWRTNumericalSemigroup
C.2-6 DeltaSetUnionUpToElementWRTNumericalSemigroup
C.2-7 DeltaSetOfNumericalSemigroup
References
Index
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