## Numerical Semigroups

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Manuel Delgado mdelgado@fc.up.pt http://www.fc.up.pt/cmup/mdelgado
Pedro A. García-Sánchez pedro@ugr.es http://www.ugr.es/~pedro
José João Morais josejoao@fc.up.pt
&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.

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.

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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Numerical Semigroups
This chapter describes how to create numerical semigroups in &GAP; and perform some basic tests.

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Basic operations with numerical semigroups
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Presentations of Numerical Semigroups
In this chapter we explain how to compute a minimal presentation of a numerical semigroup. There are three functions involved in this process.

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Constructing numerical semigroups from others
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Irreducible numerical semigroups
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Ideals of numerical semigroups
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Numerical semigroups with maximal embedding dimension
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Catenary and Tame degrees of numerical semigroups
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