  
  
               [1X[5Xnumericalsgps[105X-- a package for numerical semigroups[101X
  
  
                                 Version 1.5.0
  
  
                                 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 of the initial development stage.[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 the [13XCentro de Matemática da Universidade do
  Porto[113X.[133X
  
  [33X[0;0Y[12XFunding subsequent versions.[112X[133X
  
  [33X[0;0YVersion 1.5.x[133X
  
  [33X[0;0YBoth  maintainers  were  partially supported by partially supported by CMUP,
  member  of  LASI, which is financed by national funds through FCT – Fundação
  para  a  Ciência  e  a  Tecnologia,  I.P., under the projects with reference
  UID/00144/2025        and        associated        DOI        given       by
  [7Xhttps://doi.org/10.54499/UID/00144/2025[107X.[133X
  
  [33X[0;0YFormer versions.[133X
  
  [33X[0;0YThe   first   author   was   partially   supported   by   the   FCT  project
  PTDC/MAT/65481/2006  and  also  by the 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 grants PGC2018-096446-B-C21 and
  PID2022-138906NB-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 and
  UIDP/00144/2020.[133X
  
  [33X[0;0YBoth  maintainers  acknowledge  the  "Proyecto  de Excelencia de la Junta de
  Andalucía" (ProyExcel 00868).[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 NumberElement_NumericalSemigroup
      3.1-13 Iterator
      3.1-14 Difference
      3.1-15 AperyList
      3.1-16 AperyList
      3.1-17 AperyList
      3.1-18 AperyListOfNumericalSemigroupAsGraph
      3.1-19 KunzCoordinates
      3.1-20 KunzPolytope
      3.1-21 CocycleOfNumericalSemigroupWRTElement
      3.1-22 FrobeniusNumber
      3.1-23 Conductor
      3.1-24 PseudoFrobenius
      3.1-25 Type
      3.1-26 Gaps
      3.1-27 Weight
      3.1-28 Deserts
      3.1-29 IsOrdinary
      3.1-30 IsAcute
      3.1-31 Holes
      3.1-32 LatticePathAssociatedToNumericalSemigroup
      3.1-33 Genus
      3.1-34 FundamentalGaps
      3.1-35 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, sums, quotients, dilatations, numerical duplications
    and multiples by integers[133X
      5.2-1 Intersection
      5.2-2 \+
      5.2-3 QuotientOfNumericalSemigroup
      5.2-4 MultipleOfNumericalSemigroup
      5.2-5 NumericalDuplication
      5.2-6 AsNumericalDuplication
      5.2-7 InductiveNumericalSemigroup
      5.2-8 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 semigroups with given Frobenius
    number[133X
      5.4-1 NumericalSemigroupsWithFrobeniusNumberFG
      5.4-2 NumericalSemigroupsWithFrobeniusNumberAndMultiplicity
      5.4-3 NumericalSemigroupsWithFrobeniusNumber
      5.4-4 NumericalSemigroupsWithFrobeniusNumberPC
    5.5 [33X[0;0YConstructing the set of numerical semigroups with given maximum
    primitive[133X
      5.5-1 NumericalSemigroupsWithMaxPrimitiveAndMultiplicity
      5.5-2 NumericalSemigroupsWithMaxPrimitive
      5.5-3 NumericalSemigroupsWithMaxPrimitivePC
    5.6 [33X[0;0YConstructing the set of numerical semigroups with genus g[133X
      5.6-1 NumericalSemigroupsWithGenus
      5.6-2 NumericalSemigroupsWithGenusPC
    5.7 [33X[0;0YConstructing the set of numerical semigroups with a given set of
    pseudo-Frobenius numbers[133X
      5.7-1 ForcedIntegersForPseudoFrobenius
      5.7-2 SimpleForcedIntegersForPseudoFrobenius
      5.7-3 NumericalSemigroupsWithPseudoFrobeniusNumbers
      5.7-4 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 RemoveMinimalGeneratorFromIdeal
      7.1-5 Generators
      7.1-6 AmbientNumericalSemigroupOfIdeal
      7.1-7 IsIntegral
      7.1-8 AllIntegralIdealsContainingConductor
      7.1-9 IsComplementOfIntegralIdeal
      7.1-10 IdealByDivisorClosedSet
      7.1-11 SmallElements
      7.1-12 IdealOfNumericalSemigroupBySmallElements
      7.1-13 Conductor
      7.1-14 FrobeniusNumber
      7.1-15 PseudoFrobenius
      7.1-16 Type
      7.1-17 AddPseudoFrobeniusNumberToIdeal
      7.1-18 Minimum
      7.1-19 NormalizedIdeals
      7.1-20 BelongsToIdealOfNumericalSemigroup
      7.1-21 ElementNumber_IdealOfNumericalSemigroup
      7.1-22 NumberElement_IdealOfNumericalSemigroup
      7.1-23 \[ \]
      7.1-24 \{ \}
      7.1-25 Iterator
      7.1-26 IdealOfElementsGreaterThanOrEqualTo
      7.1-27 IsIntegrallyClosed
      7.1-28 SumIdealsOfNumericalSemigroup
      7.1-29 AsNumericalSemigroup
      7.1-30 MultipleOfIdealOfNumericalSemigroup
      7.1-31 SubtractIdealsOfNumericalSemigroup
      7.1-32 Difference
      7.1-33 TranslationOfIdealOfNumericalSemigroup
      7.1-34 IsStable
      7.1-35 IsUlrich
      7.1-36 Union
      7.1-37 Intersection
      7.1-38 MaximalIdeal
      7.1-39 CanonicalIdeal
      7.1-40 IsCanonicalIdeal
      7.1-41 IsAlmostCanonicalIdeal
      7.1-42 TraceIdeal
      7.1-43 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 KunzCoordinates
      7.3-15 KunzCoordinates
      7.3-16 AperyTable
      7.3-17 StarClosureOfIdealOfNumericalSemigroup
      7.3-18 IsReflexive
    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;0YOrder induced by a numerical semigroup[133X
    11.1 [33X[0;0YPosets induced by numerical semigroups[133X
      11.1-1 PosetNS
      11.1-2 MaximalElements
      11.1-3 MinimalElements
      11.1-4 Upset
      11.1-5 Downset
      11.1-6 AntichainsOfNumericalSemigroup
      11.1-7 Antichains
    11.2 [33X[0;0YHasse diagrams related to numerical semigroups[133X
      11.2-1 HasseDiagramOfNumericalSemigroup
      11.2-2 HasseDiagram
      11.2-3 HasseDiagramOfBettiElementsOfNumericalSemigroup
      11.2-4 HasseDiagramOfAperyListOfNumericalSemigroup
  12 [33X[0;0YAffine semigroups[133X
    12.1 [33X[0;0YDefining affine semigroups[133X
      12.1-1 AffineSemigroup
      12.1-2 AffineSemigroupByEquations
      12.1-3 AffineSemigroupByInequalities
      12.1-4 AffineSemigroupByPMInequality
      12.1-5 AffineSemigroupByGaps
      12.1-6 FiniteComplementIdealExtension
      12.1-7 Gaps
      12.1-8 Genus
      12.1-9 PseudoFrobenius
      12.1-10 SpecialGaps
      12.1-11 Generators
      12.1-12 MinimalGenerators
      12.1-13 RemoveMinimalGeneratorFromAffineSemigroup
      12.1-14 AddSpecialGapOfAffineSemigroup
      12.1-15 AsAffineSemigroup
      12.1-16 IsAffineSemigroup
      12.1-17 BelongsToAffineSemigroup
      12.1-18 IsFull
      12.1-19 HilbertBasisOfSystemOfHomogeneousEquations
      12.1-20 HilbertBasisOfSystemOfHomogeneousInequalities
      12.1-21 EquationsOfGroupGeneratedBy
      12.1-22 BasisOfGroupGivenByEquations
    12.2 [33X[0;0YGluings of affine semigroups[133X
      12.2-1 GluingOfAffineSemigroups
    12.3 [33X[0;0YPresentations of affine semigroups[133X
      12.3-1 CircuitsOfKernelCongruence
      12.3-2 PrimitiveRelationsOfKernelCongruence
      12.3-3 GeneratorsOfKernelCongruence
      12.3-4 CanonicalBasisOfKernelCongruence
      12.3-5 GraverBasis
      12.3-6 MinimalPresentation
      12.3-7 BettiElements
      12.3-8 ShadedSetOfElementInAffineSemigroup
      12.3-9 IsGeneric
      12.3-10 IsUniquelyPresented
      12.3-11 DegreesOfPrimitiveElementsOfAffineSemigroup
    12.4 [33X[0;0YFactorizations in affine semigroups[133X
      12.4-1 FactorizationsVectorWRTList
      12.4-2 Factorizations
      12.4-3 Elasticity
      12.4-4 Elasticity
      12.4-5 DeltaSet
      12.4-6 CatenaryDegree
      12.4-7 EqualCatenaryDegreeOfAffineSemigroup
      12.4-8 HomogeneousCatenaryDegreeOfAffineSemigroup
      12.4-9 MonotoneCatenaryDegreeOfAffineSemigroup
      12.4-10 TameDegree
      12.4-11 OmegaPrimality
      12.4-12 OmegaPrimality
    12.5 [33X[0;0YFinitely generated ideals of affine semigroups[133X
      12.5-1 IdealOfAffineSemigroup
      12.5-2 IsIdealOfAffineSemigroup
      12.5-3 MinimalGenerators
      12.5-4 Generators
      12.5-5 AmbientAffineSemigroupOfIdeal
      12.5-6 IsIntegral
      12.5-7 BelongsToIdealOfAffineSemigroup
      12.5-8 SumIdealsOfAffinSemigroup
      12.5-9 MultipleOfIdealOfAffineSemigroup
      12.5-10 TranslationOfIdealOfAffineSemigroup
      12.5-11 UnionIdealsOfAffineSemigroup
      12.5-12 Intersection
      12.5-13 MaximalIdeal
  13 [33X[0;0YGood semigroups[133X
    13.1 [33X[0;0YDefining good semigroups[133X
      13.1-1 IsGoodSemigroup
      13.1-2 NumericalSemigroupDuplication
      13.1-3 AmalgamationOfNumericalSemigroups
      13.1-4 CartesianProductOfNumericalSemigroups
      13.1-5 GoodSemigroup
    13.2 [33X[0;0YNotable elements[133X
      13.2-1 BelongsToGoodSemigroup
      13.2-2 Conductor
      13.2-3 Multiplicity
      13.2-4 IsLocal
      13.2-5 SmallElements
      13.2-6 RepresentsSmallElementsOfGoodSemigroup
      13.2-7 GoodSemigroupBySmallElements
      13.2-8 MaximalElementsOfGoodSemigroup
      13.2-9 IrreducibleMaximalElementsOfGoodSemigroup
      13.2-10 GoodSemigroupByMaximalElements
      13.2-11 MinimalGoodGenerators
      13.2-12 ProjectionOfAGoodSemigroup
      13.2-13 Genus
      13.2-14 Length
      13.2-15 AperySetOfGoodSemigroup
      13.2-16 StratifiedAperySetOfGoodSemigroup
    13.3 [33X[0;0YSymmetric good semigroups[133X
      13.3-1 IsSymmetric
    13.4 [33X[0;0YArf good closure[133X
      13.4-1 ArfClosure
    13.5 [33X[0;0YGood ideals[133X
      13.5-1 GoodIdeal
      13.5-2 GoodGeneratingSystemOfGoodIdeal
      13.5-3 AmbientGoodSemigroupOfGoodIdeal
      13.5-4 MinimalGoodGeneratingSystemOfGoodIdeal
      13.5-5 BelongsToGoodIdeal
      13.5-6 SmallElements
      13.5-7 CanonicalIdealOfGoodSemigroup
      13.5-8 AbsoluteIrreduciblesOfGoodSemigroup
      13.5-9 TracksOfGoodSemigroup
  14 [33X[0;0YNumerical sets[133X
    14.1 [33X[0;0YDefinitions and basic operations[133X
      14.1-1 NumericalSetBySmallElements
      14.1-2 NumericalSetByGaps
      14.1-3 AsNumericalSet
      14.1-4 AsNumericalSemigroup
      14.1-5 AsIdealOfNumericalSemigroup
      14.1-6 IsNumericalSet
      14.1-7 Difference
      14.1-8 \[ \]
      14.1-9 \{ \}
      14.1-10 Position
      14.1-11 Positions
      14.1-12 Union
      14.1-13 Intersection
      14.1-14 +
      14.1-15 *
    14.2 [33X[0;0YNotable elements of numerical sets[133X
      14.2-1 SmallElements
      14.2-2 Multiplicity
      14.2-3 Gaps
      14.2-4 Genus
      14.2-5 Conductor
      14.2-6 FrobeniusNumber
      14.2-7 DualNumericalSet
    14.3 [33X[0;0YAssociated sets to numerical semigroups[133X
      14.3-1 AtomMonoid
      14.3-2 AssociatedNumericalSets
      14.3-3 IsAssociatedNumericalSetOfNumericalSemigroup
    14.4 [33X[0;0YInteger partitions associated to numerical sets[133X
      14.4-1 IntegerPartition
      14.4-2 FerrersDiagram
      14.4-3 NumericalSetByIntegerPartition
      14.4-4 HookLengths
      14.4-5 BondedSum
      14.4-6 CojointSum
      14.4-7 EndToEndSum
  15 [33X[0;0YExternal packages[133X
    15.1 [33X[0;0YUsing external packages[133X
      15.1-1 NumSgpsUse4ti2
      15.1-2 NumSgpsUse4ti2gap
      15.1-3 NumSgpsUseNormalize
      15.1-4 NumSgpsUseSingular
      15.1-5 NumSgpsUseSingularInterface
  16 [33X[0;0YDot functions[133X
    16.1 [33X[0;0YDot functions[133X
      16.1-1 DotBinaryRelation
      16.1-2 DotTreeOfGluingsOfNumericalSemigroup
      16.1-3 DotOverSemigroupsNumericalSemigroup
      16.1-4 DotRosalesGraph
      16.1-5 DotFactorizationGraph
      16.1-6 DotEliahouGraph
      16.1-7 SetDotNSEngine
      16.1-8 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
    C.17 [33X[0;0YFunctions implemented by M. Yeşil[133X
    C.18 [33X[0;0YFunctions implemented by A. Unanue Bidal[133X
    C.19 [33X[0;0YFunctions implemented by Rajesh Kumar Sau[133X
    C.20 [33X[0;0YFunctions implemented by Neeraj Kumar[133X
  
  
  [32X
