numericalsgps-- a package for numerical semigroups Version 1.1.5 Manuel Delgado Pedro A. García-Sánchez José João Morais Manuel Delgado Email: mailto:mdelgado@fc.up.pt Homepage: http://www.fc.up.pt/cmup/mdelgado Pedro A. García-Sánchez Email: mailto:pedro@ugr.es Homepage: http://www.ugr.es/~pedro ------------------------------------------------------- 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 (http://www.fsf.org/licenses/gpl.html) 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, 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 everything 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 (NumericalSgps) 1 Introduction 2 Numerical Semigroups 2.1 Generating Numerical Semigroups 2.1-1 NumericalSemigroupByGenerators 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 GeneratorsOfNumericalSemigroup 3.1-3 EmbeddingDimension 3.1-4 SmallElements 3.1-5 FirstElementsOfNumericalSemigroup 3.1-6 RthElementOfNumericalSemigroup 3.1-7 AperyList 3.1-8 AperyList 3.1-9 AperyList 3.1-10 AperyListOfNumericalSemigroupAsGraph 3.1-11 KunzCoordinatesOfNumericalSemigroup 3.1-12 KunzPolytope 3.1-13 CocycleOfNumericalSemigroupWRTElement 3.1-14 FrobeniusNumber 3.1-15 Conductor 3.1-16 PseudoFrobeniusOfNumericalSemigroup 3.1-17 TypeOfNumericalSemigroup 3.1-18 Gaps 3.1-19 DesertsOfNumericalSemigroup 3.1-20 IsOrdinaryNumericalSemigroup 3.1-21 IsAcuteNumericalSemigroup 3.1-22 Holes 3.1-23 LatticePathAssociatedToNumericalSemigroup 3.1-24 Genus 3.1-25 FundamentalGaps 3.1-26 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 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 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 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 ANumericalSemigroupWithPseudoFrobeniusNumbers 6 Irreducible numerical semigroups 6.1 Irreducible numerical semigroups 6.1-1 IsIrreducibleNumericalSemigroup 6.1-2 IsSymmetricNumericalSemigroup 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 IsAlmostSymmetric 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 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 SumIdealsOfNumericalSemigroup 7.1-12 MultipleOfIdealOfNumericalSemigroup 7.1-13 SubtractIdealsOfNumericalSemigroup 7.1-14 Difference 7.1-15 TranslationOfIdealOfNumericalSemigroup 7.1-16 Intersection 7.1-17 MaximalIdealOfNumericalSemigroup 7.1-18 CanonicalIdealOfNumericalSemigroup 7.1-19 IsCanonicalIdeal 7.1-20 TypeSequenceOfNumericalSemigroup 7.2 Blow ups and closures 7.2-1 HilbertFunctionOfIdealOfNumericalSemigroup 7.2-2 BlowUpIdealOfNumericalSemigroup 7.2-3 ReductionNumber 7.2-4 BlowUpOfNumericalSemigroup 7.2-5 LipmanSemigroup 7.2-6 RatliffRushNumberOfIdealOfNumericalSemigroup 7.2-7 RatliffRushClosureOfIdealOfNumericalSemigroup 7.2-8 AsymptoticRatliffRushNumberOfIdealOfNumericalSemigroup 7.2-9 MultiplicitySequenceOfNumericalSemigroup 7.2-10 MicroInvariantsOfNumericalSemigroup 7.2-11 AperyListOfIdealOfNumericalSemigroupWRTElement 7.2-12 AperyTableOfNumericalSemigroup 7.2-13 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 MEDNumericalSemigroupClosure 8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup 8.2 Numerical semigroups with the Arf property and Arf closures 8.2-1 IsArf 8.2-2 ArfNumericalSemigroupClosure 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 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 FactorizationsElementListWRTNumericalSemigroup 9.1-4 RClassesOfSetOfFactorizations 9.1-5 LShapesOfNumericalSemigroup 9.1-6 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 DeltaSetPeriodicityBoundForNumericalSemigroup 9.2-8 DeltaSetPeriodicityStartForNumericalSemigroup 9.2-9 DeltaSetListUpToElementWRTNumericalSemigroup 9.2-10 DeltaSetUnionUpToElementWRTNumericalSemigroup 9.2-11 DeltaSetOfNumericalSemigroup 9.2-12 MaximumDegreeOfElementWRTNumericalSemigroup 9.2-13 MaximalDenumerantOfElementInNumericalSemigroup 9.2-14 MaximalDenumerantOfSetOfFactorizations 9.2-15 MaximalDenumerantOfNumericalSemigroup 9.2-16 AdjustmentOfNumericalSemigroup 9.2-17 IsAdditiveNumericalSemigroup 9.2-18 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 OmegaPrimalityOfElementListInNumericalSemigroup 9.4-3 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 9.6-2 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 AffineSemigroupByGenerators 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 MinimalPresentationOfAffineSemigroup 11.3-5 BettiElementsOfAffineSemigroup 11.3-6 ShadedSetOfElementInAffineSemigroup 11.3-7 IsGeneric 11.3-8 IsUniquelyPresentedAffineSemigroup 11.3-9 PrimitiveElementsOfAffineSemigroup 11.4 Factorizations in affine semigroups 11.4-1 FactorizationsVectorWRTList 11.4-2 ElasticityOfAffineSemigroup 11.4-3 DeltaSetOfAffineSemigroup 11.4-4 CatenaryDegreeOfAffineSemigroup 11.4-5 EqualCatenaryDegreeOfAffineSemigroup 11.4-6 HomogeneousCatenaryDegreeOfAffineSemigroup 11.4-7 MonotoneCatenaryDegreeOfAffineSemigroup 11.4-8 TameDegreeOfAffineSemigroup 11.4-9 OmegaPrimalityOfElementInAffineSemigroup 11.4-10 OmegaPrimalityOfAffineSemigroup 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 MinimalGoodGeneratingSystemOfGoodSemigroup 12.2-10 MinimalGenerators 12.3 Symmetric semigroups 12.3-1 IsSymmetricGoodSemigroup 12.3-2 ArfGoodSemigroupClosure 12.4 Good ideals 12.4-1 GoodIdeal 12.4-2 GoodGeneratingSystemOfGoodIdeal 12.4-3 AmbientGoodSemigroupOfGoodIdeal 12.4-4 MinimalGoodGeneratingSystemOfGoodIdeal 12.4-5 BelongsToGoodIdeal 12.4-6 SmallElementsOfGoodIdeal 12.4-7 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 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 A. Sánchez-R. Navarro C.6 Functions implemented by G. Zito C.7 Functions implemented by A. Herrera-Poyatos C.8 Functions implemented by Benjamin Heredia C.9 Functions implemented by Juan Ignacio García-García