  
  
               [1X[5Xnumericalsgps[105X-- a package for numerical semigroups[101X
  
  
                                 Version 1.3.1
  
  
                                 Manuel Delgado
  
                            Pedro A. García-Sánchez
  
                                José João Morais
  
  
  
  Manuel Delgado
      Email:    [7Xmailto:mdelgado@fc.up.pt[107X
      Homepage: [7Xhttp://www.fc.up.pt/cmup/mdelgado[107X
  Pedro A. García-Sánchez
      Email:    [7Xmailto:pedro@ugr.es[107X
      Homepage: [7Xhttp://www.ugr.es/~pedro[107X
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y©  2005--2015  Centro  de  Matemática da Universidade do Porto, Portugal and
  Universidad de Granada, Spain[133X
  
  [33X[0;0Y[13XNumericalsgps[113X  is  free  software;  you can redistribute it and/or modify it
  under     the     terms     of    the    GNU    General    Public    License
  ([7Xhttp://www.fsf.org/licenses/gpl.html[107X)  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.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThe  authors wish to thank the contributors of the package. A full list with
  the  help  received  is available in Appendix [14XC[114X. We are also in debt with H.
  Schönemann,  C.  Söeger and M. Barakat for their fruitful advices concerning
  SingularInterface, Singular, Normaliz, NormalizInterface and GradedModules.[133X
  
  [33X[0;0YThe  maintainers  want  to  thank the organizers of [10XGAPDays[110X in their several
  editions.[133X
  
  [33X[0;0YThe  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, and Santiago Melchor Ferrer for
  helping in job submission to the cluster.[133X
  
  [33X[0;0YThe  first  and second authors warmly thank María Burgos for her support and
  help.[133X
  
  [33X[0;0Y[12XFunding[112X[133X
  
  [33X[0;0YThe  first  author's  work  was  (partially)  supported  by  the  [13XCentro  de
  Matemática  da  Universidade  do  Porto[113X  (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.[133X
  
  [33X[0;0YThe   second   author  was  supported  by  the  projects  MTM2004-01446  and
  MTM2007-62346, the Junta de Andalucía group FQM-343, and FEDER founds.[133X
  
  [33X[0;0YThe third author acknowledges financial support of FCT and the POCTI program
  through  a  scholarship  given  by  [13XCentro  de Matemática da Universidade do
  Porto[113X.[133X
  
  [33X[0;0YThe   first   author   was   (partially)   supported   by  the  FCT  project
  PTDC/MAT/65481/2006  and also by the [13XCentro de Matemática da Universidade do
  Porto[113X  (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.[133X
  
  [33X[0;0YBoth  maintainers  were  (partially) supported by the projects MTM2010-15595
  and   MTM2014-55367-P,  which  were  funded  by  Ministerio  de  Economía  y
  Competitividad and the Fondo Europeo de Desarrollo Regional FEDER.[133X
  
  [33X[0;0YBoth   maintainers   want   to   acknowledge   partial   support   by   CMUP
  (UID/MAT/00144/2013   and   UID/MAT/00144/2019),  which  is  funded  by  FCT
  (Portugal)  with  national  (MEC)  and European structural funds through the
  programs FEDER, under the partnership agreement PT2020.[133X
  
  [33X[0;0YBoth   maintainers   were   also   partially   supported   by   the  project
  MTM2017-84890-P,  which is funded by Ministerio de Economía y Competitividad
  and Fondo Europeo de Desarrollo Regional FEDER.[133X
  
  [33X[0;0YThe   first   author   acknowledges   a   sabbatical  grant  from  the  FCT:
  SFRH/BSAB/142918/2018.[133X
  
  [33X[0;0YThe second author was supported in part by Grant PGC2018-096446-B-C21 funded
  by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe".[133X
  
  [33X[0;0YBoth  maintainers were partially supported by CMUP, member of LASI, which is
  financed  by Portuguese national funds through FCT – Fundação para a Ciência
  e a Tecnologia, I.P., under the project with reference UIDB/00144/2020.[133X
  
  
  -------------------------------------------------------
  [1XColophon[101X
  [33X[0;0YThis  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.[133X
  
  [33X[0;0YIf  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 [7Xhttps://www.gap-system.org/Contacts/cite.html[107X.[133X
  
  [33X[0;0YIf 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".[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (NumericalSgps)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0YNumerical Semigroups[133X
    2.1 [33X[0;0YGenerating Numerical Semigroups[133X
      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 [33X[0;0YSome basic tests[133X
      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 [33X[0;0YBasic operations with numerical semigroups[133X
    3.1 [33X[0;0YInvariants[133X
      3.1-1 Multiplicity
      3.1-2 Generators
      3.1-3 EmbeddingDimension
      3.1-4 SmallElements
      3.1-5 Length
      3.1-6 FirstElementsOfNumericalSemigroup
      3.1-7 ElementsUpTo
      3.1-8 \[ \]
      3.1-9 \{ \}
      3.1-10 NextElementOfNumericalSemigroup
      3.1-11 ElementNumber_NumericalSemigroup
      3.1-12 RthElementOfNumericalSemigroup
      3.1-13 NumberElement_NumericalSemigroup
      3.1-14 Iterator
      3.1-15 Difference
      3.1-16 AperyList
      3.1-17 AperyList
      3.1-18 AperyList
      3.1-19 AperyListOfNumericalSemigroupAsGraph
      3.1-20 KunzCoordinates
      3.1-21 KunzPolytope
      3.1-22 CocycleOfNumericalSemigroupWRTElement
      3.1-23 FrobeniusNumber
      3.1-24 Conductor
      3.1-25 PseudoFrobenius
      3.1-26 Type
      3.1-27 Gaps
      3.1-28 Weight
      3.1-29 Deserts
      3.1-30 IsOrdinary
      3.1-31 IsAcute
      3.1-32 Holes
      3.1-33 LatticePathAssociatedToNumericalSemigroup
      3.1-34 Genus
      3.1-35 FundamentalGaps
      3.1-36 SpecialGaps
    3.2 [33X[0;0YWilf's conjecture[133X
      3.2-1 WilfNumber
      3.2-2 EliahouNumber
      3.2-3 ProfileOfNumericalSemigroup
      3.2-4 EliahouSlicesOfNumericalSemigroup
  4 [33X[0;0YPresentations of Numerical Semigroups[133X
    4.1 [33X[0;0YPresentations of Numerical Semigroups[133X
      4.1-1 MinimalPresentation
      4.1-2 GraphAssociatedToElementInNumericalSemigroup
      4.1-3 BettiElements
      4.1-4 IsMinimalRelationOfNumericalSemigroup
      4.1-5 AllMinimalRelationsOfNumericalSemigroup
      4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup
      4.1-7 ShadedSetOfElementInNumericalSemigroup
    4.2 [33X[0;0YBinomial ideals associated to numerical semigroups[133X
      4.2-1 BinomialIdealOfNumericalSemigroup
    4.3 [33X[0;0YUniquely Presented Numerical Semigroups[133X
      4.3-1 IsUniquelyPresented
      4.3-2 IsGeneric
  5 [33X[0;0YConstructing numerical semigroups from others[133X
    5.1 [33X[0;0YAdding and removing elements of a numerical semigroup[133X
      5.1-1 RemoveMinimalGeneratorFromNumericalSemigroup
      5.1-2 AddSpecialGapOfNumericalSemigroup
    5.2 [33X[0;0YIntersections, quotients, dilatations, numerical duplications and
    multiples by integers[133X
      5.2-1 Intersection
      5.2-2 QuotientOfNumericalSemigroup
      5.2-3 MultipleOfNumericalSemigroup
      5.2-4 NumericalDuplication
      5.2-5 AsNumericalDuplication
      5.2-6 InductiveNumericalSemigroup
      5.2-7 DilatationOfNumericalSemigroup
    5.3 [33X[0;0YConstructing the set of all numerical semigroups containing a given
    numerical semigroup[133X
      5.3-1 OverSemigroups
    5.4 [33X[0;0YConstructing the set of numerical semigroup with given Frobenius
    number[133X
      5.4-1 NumericalSemigroupsWithFrobeniusNumberFG
      5.4-2 NumericalSemigroupsWithFrobeniusNumberAndMultiplicity
      5.4-3 NumericalSemigroupsWithFrobeniusNumber
    5.5 [33X[0;0YConstructing the set of numerical semigroups with genus g[133X
      5.5-1 NumericalSemigroupsWithGenus
    5.6 [33X[0;0YConstructing the set of numerical semigroups with a given set of
    pseudo-Frobenius numbers[133X
      5.6-1 ForcedIntegersForPseudoFrobenius
      5.6-2 SimpleForcedIntegersForPseudoFrobenius
      5.6-3 NumericalSemigroupsWithPseudoFrobeniusNumbers
      5.6-4 ANumericalSemigroupWithPseudoFrobeniusNumbers
  6 [33X[0;0YIrreducible numerical semigroups[133X
    6.1 [33X[0;0YIrreducible numerical semigroups[133X
      6.1-1 IsIrreducible
      6.1-2 IsSymmetric
      6.1-3 IsPseudoSymmetric
      6.1-4 AnIrreducibleNumericalSemigroupWithFrobeniusNumber
      6.1-5 IrreducibleNumericalSemigroupsWithFrobeniusNumber
      6.1-6 IrreducibleNumericalSemigroupsWithFrobeniusNumberAndMultiplicity
      6.1-7 DecomposeIntoIrreducibles
    6.2 [33X[0;0YComplete intersection numerical semigroups[133X
      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 IsUniversallyFree
      6.2-9 IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity
      6.2-10 NumericalSemigroupsPlanarSingularityWithFrobeniusNumber
      6.2-11 IsAperySetGammaRectangular
      6.2-12 IsAperySetBetaRectangular
      6.2-13 IsAperySetAlphaRectangular
    6.3 [33X[0;0YAlmost-symmetric numerical semigroups[133X
      6.3-1 AlmostSymmetricNumericalSemigroupsFromIrreducible
      6.3-2 AlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType
      6.3-3 IsAlmostSymmetric
      6.3-4 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber
      6.3-5 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType
    6.4 [33X[0;0YSeveral approaches generalizing the concept of symmetry[133X
      6.4-1 IsGeneralizedGorenstein
      6.4-2 IsNearlyGorenstein
      6.4-3 NearlyGorensteinVectors
      6.4-4 IsGeneralizedAlmostSymmetric
  7 [33X[0;0YIdeals of numerical semigroups[133X
    7.1 [33X[0;0YDefinitions and basic operations[133X
      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 IsComplementOfIntegralIdeal
      7.1-8 IdealByDivisorClosedSet
      7.1-9 SmallElements
      7.1-10 Conductor
      7.1-11 FrobeniusNumber
      7.1-12 PseudoFrobenius
      7.1-13 Type
      7.1-14 Minimum
      7.1-15 BelongsToIdealOfNumericalSemigroup
      7.1-16 ElementNumber_IdealOfNumericalSemigroup
      7.1-17 NumberElement_IdealOfNumericalSemigroup
      7.1-18 \[ \]
      7.1-19 \{ \}
      7.1-20 Iterator
      7.1-21 SumIdealsOfNumericalSemigroup
      7.1-22 MultipleOfIdealOfNumericalSemigroup
      7.1-23 SubtractIdealsOfNumericalSemigroup
      7.1-24 Difference
      7.1-25 TranslationOfIdealOfNumericalSemigroup
      7.1-26 Union
      7.1-27 Intersection
      7.1-28 MaximalIdeal
      7.1-29 CanonicalIdeal
      7.1-30 IsCanonicalIdeal
      7.1-31 IsAlmostCanonicalIdeal
      7.1-32 TraceIdeal
      7.1-33 TypeSequence
    7.2 [33X[0;0YDecomposition into irreducibles[133X
      7.2-1 IrreducibleZComponents
      7.2-2 DecomposeIntegralIdealIntoIrreducibles
    7.3 [33X[0;0YBlow ups and closures[133X
      7.3-1 HilbertFunctionOfIdealOfNumericalSemigroup
      7.3-2 HilbertFunction
      7.3-3 BlowUp
      7.3-4 ReductionNumber
      7.3-5 BlowUp
      7.3-6 LipmanSemigroup
      7.3-7 RatliffRushNumber
      7.3-8 RatliffRushClosure
      7.3-9 AsymptoticRatliffRushNumber
      7.3-10 MultiplicitySequence
      7.3-11 MicroInvariants
      7.3-12 AperyList
      7.3-13 AperyList
      7.3-14 AperyTable
      7.3-15 StarClosureOfIdealOfNumericalSemigroup
    7.4 [33X[0;0YPatterns for ideals[133X
      7.4-1 IsAdmissiblePattern
      7.4-2 IsStronglyAdmissiblePattern
      7.4-3 AsIdealOfNumericalSemigroup
      7.4-4 BoundForConductorOfImageOfPattern
      7.4-5 ApplyPatternToIdeal
      7.4-6 ApplyPatternToNumericalSemigroup
      7.4-7 IsAdmittedPatternByIdeal
      7.4-8 IsAdmittedPatternByNumericalSemigroup
    7.5 [33X[0;0YGraded associated ring of numerical semigroup[133X
      7.5-1 IsGradedAssociatedRingNumericalSemigroupCM
      7.5-2 IsGradedAssociatedRingNumericalSemigroupBuchsbaum
      7.5-3 TorsionOfAssociatedGradedRingNumericalSemigroup
      7.5-4 BuchsbaumNumberOfAssociatedGradedRingNumericalSemigroup
      7.5-5 IsGradedAssociatedRingNumericalSemigroupGorenstein
      7.5-6 IsGradedAssociatedRingNumericalSemigroupCI
  8 [33X[0;0YNumerical semigroups with maximal embedding dimension[133X
    8.1 [33X[0;0YNumerical semigroups with maximal embedding dimension[133X
      8.1-1 IsMED
      8.1-2 MEDClosure
      8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup
    8.2 [33X[0;0YNumerical semigroups with the Arf property and Arf closures[133X
      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
      8.2-9 ArfSpecialGaps
      8.2-10 ArfOverSemigroups
      8.2-11 IsArfIrreducible
      8.2-12 DecomposeIntoArfIrreducibles
    8.3 [33X[0;0YSaturated numerical semigroups[133X
      8.3-1 IsSaturated
      8.3-2 SaturatedClosure
      8.3-3 SaturatedNumericalSemigroupsWithFrobeniusNumber
  9 [33X[0;0YNonunique invariants for factorizations in numerical semigroups[133X
    9.1 [33X[0;0YFactorizations in Numerical Semigroups[133X
      9.1-1 FactorizationsIntegerWRTList
      9.1-2 Factorizations
      9.1-3 FactorizationsElementListWRTNumericalSemigroup
      9.1-4 RClassesOfSetOfFactorizations
      9.1-5 LShapes
      9.1-6 RFMatrices
      9.1-7 DenumerantOfElementInNumericalSemigroup
      9.1-8 DenumerantFunction
      9.1-9 DenumerantIdeal
    9.2 [33X[0;0YInvariants based on lengths[133X
      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 IsAdditiveNumericalSemigroup
      9.2-14 MaximalDenumerant
      9.2-15 MaximalDenumerantOfSetOfFactorizations
      9.2-16 MaximalDenumerant
      9.2-17 Adjustment
    9.3 [33X[0;0YInvariants based on distances[133X
      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 [33X[0;0YPrimality[133X
      9.4-1 OmegaPrimality
      9.4-2 OmegaPrimalityOfElementListInNumericalSemigroup
      9.4-3 OmegaPrimality
    9.5 [33X[0;0YHomogenization of Numerical Semigroups[133X
      9.5-1 BelongsToHomogenizationOfNumericalSemigroup
      9.5-2 FactorizationsInHomogenizationOfNumericalSemigroup
      9.5-3 HomogeneousBettiElementsOfNumericalSemigroup
      9.5-4 HomogeneousCatenaryDegreeOfNumericalSemigroup
    9.6 [33X[0;0YDivisors, posets[133X
      9.6-1 MoebiusFunctionAssociatedToNumericalSemigroup
      9.6-2 MoebiusFunction
      9.6-3 DivisorsOfElementInNumericalSemigroup
      9.6-4 NumericalSemigroupByNuSequence
      9.6-5 NumericalSemigroupByTauSequence
    9.7 [33X[0;0YFeng-Rao distances and numbers[133X
      9.7-1 FengRaoDistance
      9.7-2 FengRaoNumber
    9.8 [33X[0;0YNumerical semigroups with Apéry sets having special factorization
    properties[133X
      9.8-1 IsPure
      9.8-2 IsMpure
      9.8-3 IsHomogeneousNumericalSemigroup
      9.8-4 IsSuperSymmetricNumericalSemigroup
  10 [33X[0;0YPolynomials and numerical semigroups[133X
    10.1 [33X[0;0YGenerating functions or Hilbert series[133X
      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 CyclotomicExponentSequence
      10.1-10 WittCoefficients
      10.1-11 IsSelfReciprocalUnivariatePolynomial
    10.2 [33X[0;0YSemigroup of values of algebraic curves[133X
      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
    10.3 [33X[0;0YSemigroups and Legendrian curves[133X
      10.3-1 LegendrianGenericNumericalSemigroup
  11 [33X[0;0YAffine semigroups[133X
    11.1 [33X[0;0YDefining affine semigroups[133X
      11.1-1 AffineSemigroup
      11.1-2 AffineSemigroup
      11.1-3 AffineSemigroup
      11.1-4 AffineSemigroup
      11.1-5 AffineSemigroup
      11.1-6 Gaps
      11.1-7 Genus
      11.1-8 PseudoFrobenius
      11.1-9 SpecialGaps
      11.1-10 Generators
      11.1-11 MinimalGenerators
      11.1-12 RemoveMinimalGeneratorFromAffineSemigroup
      11.1-13 AddSpecialGapOfAffineSemigroup
      11.1-14 AsAffineSemigroup
      11.1-15 IsAffineSemigroup
      11.1-16 BelongsToAffineSemigroup
      11.1-17 IsFull
      11.1-18 HilbertBasisOfSystemOfHomogeneousEquations
      11.1-19 HilbertBasisOfSystemOfHomogeneousInequalities
      11.1-20 EquationsOfGroupGeneratedBy
      11.1-21 BasisOfGroupGivenByEquations
    11.2 [33X[0;0YGluings of affine semigroups[133X
      11.2-1 GluingOfAffineSemigroups
    11.3 [33X[0;0YPresentations of affine semigroups[133X
      11.3-1 CircuitsOfKernelCongruence
      11.3-2 PrimitiveRelationsOfKernelCongruence
      11.3-3 GeneratorsOfKernelCongruence
      11.3-4 CanonicalBasisOfKernelCongruence
      11.3-5 GraverBasis
      11.3-6 MinimalPresentation
      11.3-7 BettiElements
      11.3-8 ShadedSetOfElementInAffineSemigroup
      11.3-9 IsGeneric
      11.3-10 IsUniquelyPresented
      11.3-11 DegreesOfPrimitiveElementsOfAffineSemigroup
    11.4 [33X[0;0YFactorizations in affine semigroups[133X
      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
    11.5 [33X[0;0YFinitely generated ideals of affine semigroups[133X
      11.5-1 IdealOfAffineSemigroup
      11.5-2 IsIdealOfAffineSemigroup
      11.5-3 MinimalGenerators
      11.5-4 Generators
      11.5-5 AmbientAffineSemigroupOfIdeal
      11.5-6 IsIntegral
      11.5-7 BelongsToIdealOfAffineSemigroup
      11.5-8 SumIdealsOfAffinSemigroup
      11.5-9 MultipleOfIdealOfAffineSemigroup
      11.5-10 TranslationOfIdealOfAffineSemigroup
      11.5-11 UnionIdealsOfAffineSemigroup
      11.5-12 Intersection
      11.5-13 MaximalIdeal
  12 [33X[0;0YGood semigroups[133X
    12.1 [33X[0;0YDefining good semigroups[133X
      12.1-1 IsGoodSemigroup
      12.1-2 NumericalSemigroupDuplication
      12.1-3 AmalgamationOfNumericalSemigroups
      12.1-4 CartesianProductOfNumericalSemigroups
      12.1-5 GoodSemigroup
    12.2 [33X[0;0YNotable elements[133X
      12.2-1 BelongsToGoodSemigroup
      12.2-2 Conductor
      12.2-3 Multiplicity
      12.2-4 IsLocal
      12.2-5 SmallElements
      12.2-6 RepresentsSmallElementsOfGoodSemigroup
      12.2-7 GoodSemigroupBySmallElements
      12.2-8 MaximalElementsOfGoodSemigroup
      12.2-9 IrreducibleMaximalElementsOfGoodSemigroup
      12.2-10 GoodSemigroupByMaximalElements
      12.2-11 MinimalGoodGenerators
      12.2-12 ProjectionOfAGoodSemigroup
      12.2-13 Genus
      12.2-14 Length
      12.2-15 AperySetOfGoodSemigroup
      12.2-16 StratifiedAperySetOfGoodSemigroup
    12.3 [33X[0;0YSymmetric good semigroups[133X
      12.3-1 IsSymmetric
    12.4 [33X[0;0YArf good closure[133X
      12.4-1 ArfClosure
    12.5 [33X[0;0YGood ideals[133X
      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
      12.5-8 AbsoluteIrreduciblesOfGoodSemigroup
      12.5-9 TracksOfGoodSemigroup
  13 [33X[0;0YExternal packages[133X
    13.1 [33X[0;0YUsing external packages[133X
      13.1-1 NumSgpsUse4ti2
      13.1-2 NumSgpsUse4ti2gap
      13.1-3 NumSgpsUseNormalize
      13.1-4 NumSgpsUseSingular
      13.1-5 NumSgpsUseSingularInterface
  14 [33X[0;0YDot functions[133X
    14.1 [33X[0;0YDot functions[133X
      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 [33X[0;0YGeneralities[133X
    A.1 [33X[0;0YBézout sequences[133X
      A.1-1 BezoutSequence
      A.1-2 IsBezoutSequence
      A.1-3 CeilingOfRational
    A.2 [33X[0;0YPeriodic subadditive functions[133X
      A.2-1 RepresentsPeriodicSubAdditiveFunction
      A.2-2 IsListOfIntegersNS
  B [33X[0;0Y"Random" functions[133X
    B.1 [33X[0;0YRandom functions for numerical semigroups[133X
      B.1-1 RandomNumericalSemigroup
      B.1-2 RandomListForNS
      B.1-3 RandomModularNumericalSemigroup
      B.1-4 RandomProportionallyModularNumericalSemigroup
      B.1-5 RandomListRepresentingSubAdditiveFunction
      B.1-6 NumericalSemigroupWithRandomElementsAndFrobenius
      B.1-7 RandomNumericalSemigroupWithGenus
    B.2 [33X[0;0YRandom functions for affine semigroups[133X
      B.2-1 RandomAffineSemigroupWithGenusAndDimension
      B.2-2 RandomAffineSemigroup
      B.2-3 RandomFullAffineSemigroup
    B.3 [33X[0;0YRandom functions for good semigroups[133X
      B.3-1 RandomGoodSemigroupWithFixedMultiplicity
  C [33X[0;0YContributions[133X
    C.1 [33X[0;0YFunctions implemented by A. Sammartano[133X
    C.2 [33X[0;0YFunctions implemented by C. O'Neill[133X
    C.3 [33X[0;0YFunctions implemented by K. Stokes[133X
    C.4 [33X[0;0YFunctions implemented by I. Ojeda and C. J. Moreno Ávila[133X
    C.5 [33X[0;0YFunctions implemented by I. Ojeda[133X
    C.6 [33X[0;0YFunctions implemented by A. Sánchez-R. Navarro[133X
    C.7 [33X[0;0YFunctions implemented by G. Zito[133X
    C.8 [33X[0;0YFunctions implemented by A. Herrera-Poyatos[133X
    C.9 [33X[0;0YFunctions implemented by Benjamin Heredia[133X
    C.10 [33X[0;0YFunctions implemented by Juan Ignacio García-García[133X
    C.11 [33X[0;0YFunctions implemented by C. Cisto[133X
    C.12 [33X[0;0YFunctions implemented by N. Matsuoka[133X
    C.13 [33X[0;0YFunctions implemented by N. Maugeri[133X
    C.14 [33X[0;0YFunctions implemented by H. Martín Cruz[133X
    C.15 [33X[0;0YFunctions implemented by J. Angulo Rodríguez[133X
    C.16 [33X[0;0YFunctions implemented by F. Strazzanti[133X
  
  
  [32X
