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# Numerical Semigroups

( Version 0.95 )

Manuel Delgado
e-mail: mdelgado@fc.up.pt
WWW: http://www.fc.up.pt/cmup/mdelgado

Pedro A. García-Sánchez
e-mail: pedro@ugr.es
WWW: http://www.ugr.es/~pedro

José João Morais
e-mail: josejoao@fc.up.pt

### Copyright

(C) 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.

### 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.

### 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.

### Contents

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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