## Numerical Semigroups

( Version <#Include SYSTEM "../version"> )
Manuel Delgado mdelgado@fc.up.pt http://www.fc.up.pt/cmup/mdelgado
Pedro García-Sánchez pedro@ugr.es http://www.ugr.es/~pedro
José João Morais jjoao@netcabo.pt
&copyright; 2005 by Manuel Delgado, Pedro 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.

<#Include SYSTEM "introduction.xml">

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

<#Include SYSTEM "Generating_Numerical_Semigroups.xml"> <#Include SYSTEM "Some_basic_tests.xml">

Basic operations with numerical semigroups
<#Include SYSTEM "The_definitions.xml"> <#Include SYSTEM "Frobenius_Number.xml"> <#Include SYSTEM "Gaps.xml">

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.

<#Include SYSTEM "Presentations_of_Numerical_Semigroups.xml">

Constructing numerical semigroups from others
<#Include SYSTEM "Adding_and_removing_elements_of_a_numerical_semigroup.xml"> <#Include SYSTEM "Constructing_the_set_of_all_numerical_semigroups_containing_a_given_numerical_semigroup.xml">

Irreducible numerical semigroups
<#Include SYSTEM "Irreducible_numerical_semigroups.xml">

Ideals of numerical semigroups
<#Include SYSTEM "Ideals_of_numerical_semigroups.xml">

Numerical semigroups with maximal embedding dimension
<#Include SYSTEM "Numerical_semigroups_with_maximal_embedding_dimension.xml"> <#Include SYSTEM "Numerical_semigroups_with_the_Arf_property_and_Arf_closures.xml">

<#Include SYSTEM "generalstuff.xml">

<#Include SYSTEM "random.xml">