  
  
                              [1m[4m[31mNumerical Semigroups[0m
  
  
                                ( Version 0.95 )
  
  
                                 Manuel Delgado
  
                            Pedro A. García-Sánchez
  
                                José João Morais
  
  
  
  Manuel Delgado
      Email:    [34mmailto:mdelgado@fc.up.pt[0m
      Homepage: [34mhttp://www.fc.up.pt/cmup/mdelgado[0m
  Pedro A. García-Sánchez
      Email:    [34mmailto:pedro@ugr.es[0m
      Homepage: [34mhttp://www.ugr.es/~pedro[0m
  José João Morais
      Email:    [34mmailto:josejoao@fc.up.pt[0m
  
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  [1m[4m[31mCopyright[0m
  (C) 2005 by Manuel Delgado, Pedro A. García-Sánchez and José João Morais
  
  We  adopt  the  copyright  regulations  of  [1mGAP[0m as detailed in the copyright
  notice in the [1mGAP[0m manual.
  
  
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  [1m[4m[31mAcknowledgements[0m
  The  first  author's  work  was  (partially)  supported  by  the  [22m[36mCentro  de
  Matemática  da  Universidade  do  Porto[0m  (CMUP),  financed by FCT (Portugal)
  through  the  programmes  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 project MTM2004-01446 and FEDER
  founds.
  
  The third author acknowledges financial support of FCT and the POCTI program
  through  a  scholarship  given  by  [22m[36mCentro  de Matemática da Universidade do
  Porto[0m.
  
  The authors whish to thank J. I. García-García for many helpfull discussions
  and  for  helping  in  the  programming  of  preliminary  versions  of  some
  functions.
  
  
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  [1m[4m[31mColophon[0m
  This work started when the first author visited the University of Granada in
  part  of  a  sabbatical  year. Bug reports, suggestions and comments are, of
  course, welcome. Please use our email addresses to this effect.
  
  
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  [1m[4m[31mContent (NumericalSgps)[0m
  
  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 SmallElementsOfNumericalSemigroup
      3.1-4 FirstElementsOfNumericalSemigroup
      3.1-5 AperyListOfNumericalSemigroupWRTElement
      3.1-6 DrawAperyListOfNumericalSemigroup
      3.1-7 AperyListOfNumericalSemigroupAsGraph
    3.2 Frobenius Number
      3.2-1 FrobeniusNumberOfNumericalSemigroup
      3.2-2 FrobeniusNumber
      3.2-3 PseudoFrobeniusOfNumericalSemigroup
    3.3 Gaps
      3.3-1 GapsOfNumericalSemigroup
      3.3-2 FundamentalGapsOfNumericalSemigroup
      3.3-3 SpecialGapsOfNumericalSemigroup
  4. Presentations of Numerical Semigroups
    4.1 Presentations of Numerical Semigroups
      4.1-1 FortenTruncatedNCForNumericalSemigroups
      4.1-2 MinimalPresentationOfNumericalSemigroup
      4.1-3 GraphAssociatedToElementInNumericalSemigroup
  5. Constructing numerical semigroups from others
    5.1 Adding and removing elements of a numerical semigroup
      5.1-1 RemoveMinimalGeneratorFromNumericalSemigroup
      5.1-2 AddSpecialGapOfNumericalSemigroup
      5.1-3 IntersectionOfNumericalSemigroups
      5.1-4 QuotientOfNumericalSemigroup
    5.2 Constructing the set of all numerical semigroups containing a given
    numerical semigroup
      5.2-1 OverSemigroupsNumericalSemigroup
      5.2-2 NumericalSemigroupsWithFrobeniusNumber
  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
  7. Ideals of numerical semigroups
    7.1 Ideals of numerical semigroups
      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 HilbertFunctionOfIdealOfNumericalSemigroup
      7.1-14 BlowUpIdealOfNumericalSemigroup
      7.1-15 ReductionNumberIdealNumericalSemigroup
      7.1-16 MaximalIdealOfNumericalSemigroup
      7.1-17 BlowUpOfNumericalSemigroup
      7.1-18 MicroInvariantsOfNumericalSemigroup
      7.1-19 IsGradedAssociatedRingNumericalSemigroupCM
      7.1-20 CanonicalIdealOfNumericalSemigroup
      7.1-21 IntersectionIdealsOfNumericalSemigroup
  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
  9. Catenary and Tame degrees of numerical semigroups
    9.1 Factorizations in Numerical Semigroups
      9.1-1 FactorizationsElementWRTNumericalSemigroup
      9.1-2 LengthsOfFactorizationsElementWRTNumericalSemigroup
      9.1-3 ElasticityOfFactorizationsElementWRTNumericalSemigroup
      9.1-4 ElasticityOfNumericalSemigroup
      9.1-5 DeltaSetOfFactorizationsElementWRTNumericalSemigroup
      9.1-6 MaximumDegreeOfElementWRTNumericalSemigroup
      9.1-7 CatenaryDegreeNumericalSemigroup
      9.1-8 TameDegreeNumericalSemigroup
  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
  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. A graphical interface
    C.1 Graphical interface
      C.1-1 XNumericalSemigroup
  
  
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