Numerical Semigroups
  
  
                                ( Version 0.96 )
  
  
                                 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
  José João Morais
      Email:    mailto:josejoao@fc.up.pt
  
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  Copyright
  © 2005 by Manuel Delgado, Pedro A. García-Sánchez and José João Morais
  
  We  adopt  the  copyright  regulations  of  GAP as detailed in the copyright
  notice in the GAP manual.
  
  
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  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  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  Centro  de Matemática da Universidade do
  Porto.
  
  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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  Colophon
  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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  Contents (NumericalSgps)
  
  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
      5.2-3 NumericalSemigroupsWithGenus
  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
      7.1-22 IsMonomialNumericalSemigroup
  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 CatenaryDegreeOfNumericalSemigroup
      9.1-8 CatenaryDegreeOfElementNS
      9.1-9 TameDegreeOfNumericalSemigroup
  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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