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<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE Book SYSTEM "gapdoc.dtd">

<Book Name="NumericalSgps">

<TitlePage>
 <Title><Package>numericalsgps</Package>-- a package for numerical semigroups</Title>
 <Version> Version <#Include SYSTEM "../version">
 </Version>
 <Author> Manuel Delgado
 <Email>mdelgado@fc.up.pt</Email>
 <Homepage>http://www.fc.up.pt/cmup/mdelgado</Homepage>
 </Author>
 <Author> Pedro A. García-Sánchez
 <Email>pedro@ugr.es</Email>
 <Homepage>http://www.ugr.es/~pedro</Homepage>
 </Author>
 <Author> José João Morais

 </Author>
 <Copyright> ©right; 2005--2015 Centro de Matemática da Universidade do Porto, Portugal and Universidad de Granada, Spain
 
<E>Numericalsgps</E> is free software; you can redistribute it and/or modify it under the terms of the
<URL Text="GNU General Public License">http://www.fsf.org/licenses/gpl.html</URL> as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the file 'GPL' included in the package or see the FSF's own site.
</Copyright>

<Acknowledgements>

The authors wish to thank the contributors of the package. A full list with the help received is available in Appendix <Ref Chap="ap:contrib"/>. We are also in debt with H. Schönemann, C. Söeger and M. Barakat for their fruitful advices concerning SingularInterface, Singular, Normaliz, NormalizInterface and GradedModules.
 The maintainers want to thank the organizers of <C>GAPDays</C> in their several editions.
 The authors also thank the Centro de Servicios de Informática y Redes de Comunicaciones (CSIRC), Universidad de Granada, for providing the computing time, specially Rafael Arco Arredondo for installing this package and the extra software needed in alhambra.ugr.es, and Santiago Melchor Ferrer for helping in job submission to the cluster.

The first and second authors warmly thank María Burgos for her support and help.


Funding


The first author's work was (partially) supported by the Centro de Matemática da Universidade do
Porto</E> (CMUP), financed by FCT (Portugal) through the programs 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 projects MTM2004-01446 and MTM2007-62346, the Junta de Andalucía group FQM-343, and FEDER founds.

The third author acknowledges financial support of FCT and the POCTI program
through a scholarship given by <E>Centro de Matemática da Universidade do Porto</E>.

The first author was (partially) supported by the FCT project PTDC/MAT/65481/2006 and also by the
<E>Centro de Matemática da Universidade do
Porto</E> (CMUP), funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011.

Both maintainers were (partially) supported by the projects MTM2010-15595 and MTM2014-55367-P, which were funded by Ministerio de Economía y Competitividad and the Fondo Europeo de Desarrollo Regional FEDER.
 Both maintainers want to acknowledge partial support by CMUP (UID/MAT/00144/2013 and UID/MAT/00144/2019), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.
 Both maintainers were also partially supported by the project MTM2017-84890-P, which is funded by Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional FEDER.
 The first author acknowledges a sabbatical grant from the FCT: SFRH/BSAB/142918/2018.

The second author was supported in part by Grant PGC2018-096446-B-C21 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe".

Both maintainers were partially supported by CMUP, member of LASI, which is financed by Portuguese national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020 and UIDP/00144/2020.

Both maintainers acknowledge the "Proyecto de Excelencia de la Junta de Andalucía" (ProyExcel 00868).
</Acknowledgements>

<Colophon>
This work started when (in 2004) the first author visited the University of Granada in part of a sabbatical year.

Since Version 0.96 (released in 2008), the package is maintained by the first two authors.

Bug reports, suggestions and comments are, of course, welcome. Please use our
email addresses to this effect.


If you have benefited from the use of the numerigalsgps GAP package in your research, please cite it in addition to GAP itself, following the scheme proposed in
<URL>https://www.gap-system.org/Contacts/cite.html</URL>.


If you have predominantly used the functions in the Appendix, contributed by other authors, please cite in addition these authors, referring "software implementations available in the GAP package NumericalSgps".
</Colophon>

</TitlePage>

<TableOfContents/>

<Body>

<#Include SYSTEM "introduction.xml">

<Chapter>
 <Heading>
 Numerical Semigroups
 </Heading>
 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">

</Chapter>

 <Chapter>
 <Heading>
 Basic operations with numerical semigroups
 </Heading>
 This chapter describes some basic functions to deal with notable elements in a numerical semigroup. A section including functions to test Wilf's conjecture is also included in this chapter. We provide some functions that allow to treat a numerical semigroup as a list, and thus easy the task to access to its elements.

<#Include SYSTEM "The_definitions.xml">
<#Include SYSTEM "Wilf.xml">

</Chapter>

<Chapter Label="ch:min-pres">

 <Heading>
 Presentations of Numerical Semigroups
 </Heading>
 In this chapter we explain how to compute a minimal presentation of a numerical
 semigroup. Recall that a minimal presentation is a minimal generating system of the kernel congruence of the factorization map of the numerical semigroup. If <M>S</M> is a numerical semigroup minimally generated by <M>\{n_1,\ldots,n_e\}</M>, then the factorization map is the epimorphism <M>\varphi: \mathbb{N}^e\to S</M>, <M>(x_1,\ldots,x_e)\mapsto x_1n_1+\dots+ x_en_e</M>; its kernel is the congruence <M>\{ (a,b) \mid \varphi(a)=\varphi(b)\}</M>.
 

 The set of minimal generators is stored in a set, and so it may not be arranged as the user gave them. This may affect the arrangement of the coordinates of the pairs in a minimal presentation, since every coordinate is associated to a minimal generator.

<#Include SYSTEM "Presentations_of_Numerical_Semigroups.xml">

</Chapter>

 <Chapter>
 <Heading>
 Constructing numerical semigroups from others
 </Heading>
 This chapter provides several functions to construct numerical semigroups from others (via intersections, quotients by an integer, removing or adding integers, etc.).

<#Include SYSTEM "Adding_and_removing_elements_of_a_numerical_semigroup.xml">
<#Include SYSTEM "Operations_Numerical_Semigroups.xml">
<#Include SYSTEM "Constructing_sets_of_numerical_semigroups.xml">

</Chapter>

 <Chapter>
 <Heading>
 Irreducible numerical semigroups
 </Heading>
 An irreducible numerical semigroup is a semigroup that cannot be expressed as the
 intersection of numerical semigroups properly containing it.
 

 It is not difficult to prove that a semigroup is irreducible if and only if it is
 maximal (with respect to set inclusion) in the set of all numerical semigroups having
 its same Frobenius number (see <Cite Key="RB03"></Cite>). Hence, according to
 <Cite Key="FGH87"></Cite> (respectively <Cite Key="BDF97"></Cite>),
 symmetric (respectively pseudo-symmetric) numerical semigroups are those irreducible
 numerical semigroups with odd (respectively even) Frobenius number.
 

 In <Cite Key="RGGJ03"></Cite> it is shown that a nontrivial numerical semigroup is irreducible if and
 only if it has only one special gap. We use this characterization.
 

 In old versions of the package, we first constructed an irreducible numerical semigroup with
 the given Frobenius number (as explained in <Cite Key="RGS04"></Cite>), and then
 we constructed the rest from it.
 The present version uses a faster procedure presented in <Cite Key="BR13"></Cite>.
 

 Every numerical semigroup can be expressed as an intersection of irreducible
 numerical semigroups. If <M>S</M> can be expressed as <M>S=S_1\cap \cdots\cap S_n</M>,
 with <M>S_i</M> irreducible numerical semigroups, and no factor can be removed, then
 we say that this decomposition is minimal. Minimal decompositions can be computed by
 using Algorithm 26 in <Cite Key="RGGJ03"></Cite>.
 

<#Include SYSTEM "Irreducible_numerical_semigroups.xml">
<#Include SYSTEM "Complete_Intersections.xml">
<#Include SYSTEM "Almost_symmetric.xml">
<#Include SYSTEM "generalized_symmetric.xml">
</Chapter>

 <Chapter>
 <Heading>
 Ideals of numerical semigroups
 </Heading>

 Let <M>S</M> be a numerical semigroup. A set <M>I</M> of integers is an <E>ideal relative</E>
 to a numerical semigroup <M>S</M> provided that <M>I+S\subseteq I</M> and that there
 exists <M>d\in S</M> such that <M>d+I\subseteq S</M>.
 
 If <M>\{i_1,\ldots,i_k\}</M> is a subset of <M>{\mathbb Z}</M>, then the set
 <M>I=\{i_1,\ldots,i_k\}+S=\bigcup_{n=1}^k i_n+S</M> is an ideal relative to <M>S</M>,
 and <M>\{i_1,\ldots, i_k\}</M> is a system of generators of <M>I</M>. A system of generators
 <M>M</M> is minimal if no proper subset of <M>M</M> generates the same ideal.
 Usually, ideals are specified by means of its generators and the ambient numerical
 semigroup to which they are ideals (for more information see for instance
 <Cite Key="BDF97"></Cite>).

<#Include SYSTEM "Ideals_of_numerical_semigroups.xml">

</Chapter>

 <Chapter>
 <Heading>
 Numerical semigroups with maximal embedding dimension
 </Heading>

 If <M>S</M> is a numerical semigroup and <M>m</M> is its multiplicity (the least
 positive integer belonging to it), then the embedding dimension <M>e</M> of <M>S</M> (the
 cardinality of the minimal system of generators of <M>S</M>) is less than or equal
 to <M>m</M>. We say that <M>S</M> has <E>maximal embedding dimension</E> (MED for short) when <M>e=m</M>.
 The intersection of two numerical semigroups with the same multiplicity and maximal embedding
 dimension is again of maximal embedding dimension. Thus we define the MED closure of a non-empty subset
 of positive integers <M>M=\{m < m_1 < \cdots < m_n <\cdots\}</M> with <M>\gcd(M)=1</M>
 as the intersection of all MED numerical semigroups with multiplicity <M>m</M>.

 
 Given a MED numerical semigroup <M>S</M>, we say that <M>M=\{m_1 < \cdots< m_k\}</M> is a MED system
 of generators if the MED closure of <M>M</M> is <M>S</M>. Moreover, <M>M</M> is a minimal MED generating
 system for <M>S</M> provided that every proper subset of <M>M</M> is not a MED system of generators
 of <M>S</M>. Minimal MED generating systems are unique, and in general are smaller than the
 classical minimal generating systems (see <Cite Key="RGGB03"></Cite>).

<#Include SYSTEM "Numerical_semigroups_with_maximal_embedding_dimension.xml">
</Chapter>

<Chapter Label="ch:factorizations">
<Heading>
 Nonunique invariants for factorizations in numerical semigroups
</Heading>
 Let <M> S </M> be a numerical semigroup minimally generated by
 <M> \{m_1,\ldots,m_n\} </M>. A <E>factorization</E> of an element <M>s\in S</M>
 is an n-tuple <M> a=(a_1,\ldots,a_n) </M> of nonnegative integers such that
 <M> n=a_1 n_1+\cdots+a_n m_n</M>. The <E>length</E> of <M>a</M> is <M>|a|=a_1+\cdots+a_n</M>.
 Given two factorizations <M>a</M> and <M>b</M> of <M>n</M>, the <E>distance</E>
 between <M>a</M> and <M>b</M> is <M>d(a,b)=\max \{ |a-\gcd(a,b)|,|b-\gcd(a,b)|\}</M>,
 where <M>\gcd((a_1,\ldots,a_n),(b_1,\ldots,b_n))=(\min(a_1,b_1),\ldots,\min(a_n,b_n))</M>.
 In the literature, factorizations are sometimes called representations or expressions of the
 element in terms of the generators.

 
 If <M>l_1>\cdots > l_k</M> are the lengths of all the factorizations of
 <M>s \in S</M>, the <E>delta set</E> associated to <M>s</M> is
 <M>\Delta(s)=\{l_1-l_2,\ldots,l_k-l_{k-1}\}</M>.

 
 The <E>catenary degree</E> of an element in <M>S</M> is the least positive integer <M>c</M> such that for any two
 of its factorizations <M>a</M> and <M>b</M>, there exists a chain of
 factorizations starting in <M>a</M> and ending in <M>b</M> and so that the distance between
 two consecutive links is at most <M>c</M>. The <E>catenary degree</E> of <M>S</M> is the supremum of the
 catenary degrees of the elements in <M>S</M>.

 
 The <E>tame degree</E> of <M>S</M> is the least positive integer <M>t</M> such that for any
 factorization <M>a</M> of an element <M>s</M> in <M>S</M>, and any <M>i</M> such that
 <M>s-m_i\in S</M>, there exists another factorization <M>b</M> of <M>s</M> so that
 the distance to <M>a</M> is at most <M>t</M> and <M>b_i\not = 0</M>.


The <E><M>\omega</M>-primality</E> of an element <M>s</M> in <M>S</M> is the least positive integer <M>k</M> such that if
<M>(\sum_{i\in I} s_i)-s\in S, s_i\in S</M>, then there exists <M>\Omega\subseteq I</M> with cardinality <M>k</M> such that
<M>(\sum_{i\in \Omega} s_i)-s\in S</M>. The <E><M>\omega</M>-primality</E> of <M>S</M> is the maximum of the <M>\omega</M>-primality of its minimal
generators.

 
 The basic properties of these constants can be found in <Cite Key="GHKb"></Cite>.
 The algorithm used to compute the catenary and tame degree
 is an adaptation of the algorithms appearing
 in <Cite Key="CGLPR"></Cite> for numerical semigroups (see <Cite Key="CGL"></Cite>).
 The computation of the elasticity of a numerical semigroup reduces to <M>m/n</M> with <M>m</M> the
 multiplicity of the semigroup and <M>n</M> its largest minimal generator (see <Cite Key="CHM06"></Cite>
 or <Cite Key="GHKb"></Cite>).

<#Include SYSTEM "catenary-tame.xml">

</Chapter>

<Chapter>

<Heading>
 Polynomials and numerical semigroups
</Heading>

 Polynomials appear related to numerical semigroups in several ways. One of them is through their associated generating function (or Hilbert series), and another via value semigroups of a curve; and curves might be defined by polynomials. In this chapter we present several functions to compute the polynomial and Hilbert series associated to a numerical semigroup, and to calculate the respective numerical semigroups given a set of defining polynomials.

<#Include SYSTEM "polynomial.xml">

</Chapter>

<Chapter>
<Heading>
 Affine semigroups
</Heading>

 An <E>affine semigroup</E> <M>S</M> is a finitely generated cancellative monoid that is reduced (no units other than 0) and is torsion-free (<M> a s= b s</M> implies <M>a=b</M>, with <M>a,b\in \mathbb N</M> and <M>s\in S</M>).

 Up to isomorphism any affine semigroup can be viewed as a finitely generated submonoid of <M>\mathbb N^k</M> for some positive integer <M>k</M>. Thus affine semigroups are a natural generalization of numerical semigroups.
 
 Some of the functions in this chapter may work considerably faster when some external package is installed and its use is allowed. When this is the case, it is referred in the function documentation. We refer the user to Chapter <Ref Chap="ch:external"/> for details on the use of external packages.

<#Include SYSTEM "affine.xml">

</Chapter>

<Chapter>
<Heading>
 Good semigroups
</Heading>

We will only cover here good semigroups of <M>\mathbb{N}^2</M>. 

A <E>good semigroup</E> <M>S</M> is a submonoid of <M>\mathbb{N}^2</M>, with the following properties. 
(G1) It is closed under infimums (minimum componentwise). 
(G2) If <M>a, b \in M</M> and <M>a_i = b_i</M> for some <M>i \in \{1, 2\}</M>, then there exists <M>c \in M</M> such that <M>c_i > a_i = b_i</M>
and <M>c_j = \min\{a_j,b_j\}</M>, with <M>j∈\{1,2\}\setminus \{i\}</M>.
(G3) There exists <M>C\in\mathbb{N}^n</M> such that <M>C+\mathbb{N}^n\subseteq S</M>. 

Value semigroups of algebroid branches are good semigroups, but there are good semigroups that are not of this form. Since good semigroups are closed under infimums, if <M>C_1</M> and <M>C_2</M> fulfill <M>C_i+\mathbb{N}^n\subseteq S</M>, then <M>C_1\wedge C_2+\mathbb{N}^n\subseteq S</M>. So there is a minimum <M>C</M> fulfilling <M>C+\mathbb{N}^n\subseteq S</M>, which is called the <E>conductor</E> of <M>S</M>. 

The contents of this chapter are described in <Cite Key="DGSM"></Cite>.

<#Include SYSTEM "good-semigroups.xml">

</Chapter>

<Chapter Label="ch:external">
<Heading>
 External packages
</Heading>

<#Include SYSTEM "external-packages.xml">

</Chapter>

<Chapter Label="ch:dot">
<Heading>
 Dot functions
</Heading>

<#Include SYSTEM "dot.xml">
</Chapter>

</Body>

<#Include SYSTEM "generalstuff.xml">

<#Include SYSTEM "random.xml">

<#Include SYSTEM "contributions.xml">

<Bibliography Databases="NumericalSgps" />

<TheIndex/>

</Book>

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