[1XNumerical Semigroups[0m ( Version 0.96 ) Manuel Delgado Pedro A. GarcÃa-Sánchez José João Morais Manuel Delgado Email: [7Xmailto:mdelgado@fc.up.pt[0m Homepage: [7Xhttp://www.fc.up.pt/cmup/mdelgado[0m Pedro A. GarcÃa-Sánchez Email: [7Xmailto:pedro@ugr.es[0m Homepage: [7Xhttp://www.ugr.es/~pedro[0m José João Morais Email: [7Xmailto:josejoao@fc.up.pt[0m ------------------------------------------------------- [1XCopyright[0m © 2005 by Manuel Delgado, Pedro A. GarcÃa-Sánchez and José João Morais We adopt the copyright regulations of [5XGAP[0m as detailed in the copyright notice in the [5XGAP[0m manual. ------------------------------------------------------- [1XAcknowledgements[0m The first author's work was (partially) supported by the [13XCentro 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 [13XCentro de Matemática da Universidade do Porto[0m. The authors whish to thank J. I. GarcÃa-GarcÃa for many helpfull discussions and for helping in the programming of preliminary versions of some functions. ------------------------------------------------------- [1XColophon[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. ------------------------------------------------------- [1XContents (NumericalSgps)[0X 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 -------------------------------------------------------