  
  
               [1X[5Xnumericalsgps[105X-- a package for numerical semigroups[101X
  
  
                                 Version 1.0.1
  
  
                                 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.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [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, 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 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.[133X
  
  [33X[0;0YThe  first  and second authors warmly thank María Burgos for her support and
  help.[133X
  
  [33X[0;0Y[12XConcerning the mantainment:[112X[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;0YThe  second  author  was/is  supported  by  the  projects  MTM2007-62346 and
  MTM2010-15595.[133X
  
  [33X[0;0YBoth    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.[133X
  
  [33X[0;0YThe  maintainers  want  to  thank the organizers of [10XGAPDays[110X in their several
  editions.[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 [7Xhttp://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 ModularNumericalSemigroup
      2.1-3 ProportionallyModularNumericalSemigroup
      2.1-4 NumericalSemigroupByGenerators
    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 BelongsToNumericalSemigroup
  3 [33X[0;0YBasic operations with numerical semigroups[133X
    3.1 [33X[0;0YThe definitions[133X
      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 [33X[0;0YFrobenius Number[133X
      3.2-1 FrobeniusNumberOfNumericalSemigroup
      3.2-2 FrobeniusNumber
      3.2-3 ConductorOfNumericalSemigroup
      3.2-4 PseudoFrobeniusOfNumericalSemigroup
      3.2-5 TypeOfNumericalSemigroup
    3.3 [33X[0;0YGaps[133X
      3.3-1 GapsOfNumericalSemigroup
      3.3-2 GenusOfNumericalSemigroup
      3.3-3 FundamentalGapsOfNumericalSemigroup
      3.3-4 SpecialGapsOfNumericalSemigroup
  4 [33X[0;0YPresentations of Numerical Semigroups[133X
    4.1 [33X[0;0YPresentations of Numerical Semigroups[133X
      4.1-1 MinimalPresentationOfNumericalSemigroup
      4.1-2 GraphAssociatedToElementInNumericalSemigroup
      4.1-3 BettiElementsOfNumericalSemigroup
      4.1-4 PrimitiveElementsOfNumericalSemigroup
      4.1-5 ShadedSetOfElementInNumericalSemigroup
    4.2 [33X[0;0YUniquely Presented Numerical Semigroups[133X
      4.2-1 IsUniquelyPresentedNumericalSemigroup
      4.2-2 IsGenericNumericalSemigroup
  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 and quotients by integers[133X
      5.2-1 IntersectionOfNumericalSemigroups
      5.2-2 QuotientOfNumericalSemigroup
    5.3 [33X[0;0YConstructing the set of all numerical semigroups containing a given
    numerical semigroup[133X
      5.3-1 OverSemigroupsNumericalSemigroup
    5.4 [33X[0;0YConstructing the set of numerical semigroup with given Frobenius
    number[133X
      5.4-1 NumericalSemigroupsWithFrobeniusNumber
    5.5 [33X[0;0YConstructing the set of numerical semigroups with genus g, that is,
    numerical semigroups with exactly g gaps[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 RandomNumericalSemigroupWithPseudoFrobeniusNumbers
  6 [33X[0;0YIrreducible numerical semigroups[133X
    6.1 [33X[0;0YIrreducible numerical semigroups[133X
      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 [33X[0;0YComplete intersection numerical semigroups[133X
      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 [33X[0;0YAlmost-symmetric numerical semigroups[133X
      6.3-1 AlmostSymmetricNumericalSemigroupsFromIrreducible
      6.3-2 IsAlmostSymmetricNumericalSemigroup
      6.3-3 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber
  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 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 [33X[0;0YOther functions for ideals[133X
      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 [33X[0;0YNumerical semigroups with maximal embedding dimension[133X
    8.1 [33X[0;0YNumerical semigroups with maximal embedding dimension[133X
      8.1-1 IsMEDNumericalSemigroup
      8.1-2 MEDNumericalSemigroupClosure
      8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup
    8.2 [33X[0;0YNumerical semigroups with the Arf property and Arf closures[133X
      8.2-1 IsArfNumericalSemigroup
      8.2-2 ArfNumericalSemigroupClosure
      8.2-3 MinimalArfGeneratingSystemOfArfNumericalSemigroup
      8.2-4 ArfNumericalSemigroupsWithFrobeniusNumber
    8.3 [33X[0;0YSaturated numerical semigroups[133X
      8.3-1 IsSaturatedNumericalSemigroup
      8.3-2 SaturatedNumericalSemigroupClosure
      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 FactorizationsElementWRTNumericalSemigroup
      9.1-3 RClassesOfSetOfFactorizations
      9.1-4 LShapesOfNumericalSemigroup
      9.1-5 DenumerantOfElementInNumericalSemigroup
    9.2 [33X[0;0YInvariants based on lengths[133X
      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 [33X[0;0YInvariants based on distances[133X
      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 [33X[0;0YPrimality[133X
      9.4-1 OmegaPrimalityOfElementInNumericalSemigroup
      9.4-2 OmegaPrimalityOfNumericalSemigroup
    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
  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 HilbertSeriesOfNumericalSemigroup
      10.1-3 GraeffePolynomial
      10.1-4 IsCyclotomicPolynomial
      10.1-5 IsKroneckerPolynomial
      10.1-6 IsCyclotomicNumericalSemigroup
      10.1-7 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 SemigroupOfValuesOfCurve_Local
      10.2-6 SemigroupOfValuesOfCurve_Global
  11 [33X[0;0YAffine semigroups[133X
    11.1 [33X[0;0YUsing external packages[133X
      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 [33X[0;0YDefining affine semigroups[133X
      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 [33X[0;0YGluings of affine semigroups[133X
      11.3-1 GluingOfAffineSemigroups
    11.4 [33X[0;0YPresentations of affine semigroups[133X
      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 [33X[0;0YFactorizations in affine semigroups[133X
      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 [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;0YRandom functions[133X
    B.1 [33X[0;0YRandom functions[133X
      B.1-1 RandomNumericalSemigroup
      B.1-2 RandomListForNS
      B.1-3 RandomModularNumericalSemigroup
      B.1-4 RandomProportionallyModularNumericalSemigroup
      B.1-5 RandomListRepresentingSubAdditiveFunction
  C [33X[0;0YContributions[133X
    C.1 [33X[0;0YFunctions implemented by A. Sammartano[133X
      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 [33X[0;0YFunctions implemented by C. O'Neill[133X
      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
  
  
  [32X
