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numericalsgps-- a package for numerical semigroups

Version 1.0.1

Manuel Delgado

Pedro A. García-Sánchez

José João Morais


© 2005--2015 Centro de Matemática da Universidade do Porto, Portugal and Universidad de Granada, Spain

Numericalsgps is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.


The first author's work was (partially) supported by the Centro de Matemática da Universidade do Porto (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, FQM-343, 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 wish to thank J. I. García-García and Alfredo Sánchez-R. Navarro for many helpful discussions and for helping in the programming of preliminary versions of some functions, and also to C. O'Neill and A. Sammartano for their contributions (see Contributions Chapter). We are also in debt with S. Gutsche, M. Horn, H. Schönemann, C. Söeger and M. Barakat for their fruitful advices concerning 4ti2Interface, SingularInterface, Singular, Normaliz, NormalizInterface and GradedModules.

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

Concerning the mantainment:

The first author was (partially) supported by the FCT project PTDC/MAT/65481/2006 and also by the Centro de Matemática da Universidade do Porto (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.

The second author was/is supported by the projects MTM2007-62346 and MTM2010-15595.

Both authors want to acknowledge partial support by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.

The maintainers want to thank the organizers of GAPDays in their several editions.


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

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".


1 Introduction
2 Numerical Semigroups
3 Basic operations with numerical semigroups
4 Presentations of Numerical Semigroups
5 Constructing numerical semigroups from others
6 Irreducible numerical semigroups
7 Ideals of numerical semigroups
8 Numerical semigroups with maximal embedding dimension
9 Nonunique invariants for factorizations in numerical semigroups
10 Polynomials and numerical semigroups
11 Affine semigroups
A Generalities
B Random functions
C Contributions

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