[MathJax on]

James Mitchell
Email: jdm3@st-andrews.ac.uk
Homepage: https://jdbm.me
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Marina Anagnostopoulou-Merkouri
Email: mam49@st-andrews.ac.uk
Homepage: https://github.com/marinaanagno
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Thomas Breuer
Email: sam@math.rwth-aachen.de
Homepage: https://www.math.rwth-aachen.de/~Thomas.Breuer/

Stuart Burrell
Email: stuartburrell1994@gmail.com
Homepage: https://stuartburrell.github.io

Reinis Cirpons
Email: rc234@st-andrews.ac.uk
Homepage: https://reinisc.id.lv/
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Tom Conti-Leslie
Email: tom.contileslie@gmail.com
Homepage: https://tomcontileslie.com/

Joseph Edwards
Email: jde1@st-andrews.ac.uk
Homepage: https://github.com/Joseph-Edwards
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Attila Egri-Nagy
Email: attila@egri-nagy.hu
Homepage: http://www.egri-nagy.hu

Luke Elliott
Email: le27@st-andrews.ac.uk
Homepage: https://le27.github.io/Luke-Elliott/
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Fernando Flores Brito
Email: ffloresbrito@gmail.com
Homepage: https://github.com/ffloresbrito

Nick Ham
Email: nicholas.charles.ham@gmail.com
Homepage: https://n-ham.github.io

Robert Hancock
Email: hancock@informatik.uni-heidelberg.de
Homepage: https://sites.google.com/view/robert-hancock/

Max Horn
Email: horn@mathematik.uni-kl.de
Homepage: https://www.quendi.de/math
Address:
Fachbereich Mathematik, TU Kaiserslautern, Gottlieb-Daimler-Straße 48, 67663 Kaiserslautern, Germany

Christopher Jefferson
Email: caj21@st-andrews.ac.uk
Homepage: https://caj.host.cs.st-andrews.ac.uk/
Address:
Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland

Julius Jonusas
Email: j.jonusas@gmail.com
Homepage: http://julius.jonusas.work

Chinmaya Nagpal
Email: chinmaya1011@gmail.com
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Olexandr Konovalov
Email: obk1@st-andrews.ac.uk
Homepage: https://olexandr-konovalov.github.io/
Address:
Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland

Artemis Konstantinidi
Email: ak316@st-andrews.ac.uk
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Dima V. Pasechnik
Email: dmitrii.pasechnik@cs.ox.ac.uk
Homepage: http://users.ox.ac.uk/~coml0531/
Address:
Pembroke College, St. Aldates, Oxford OX1 1DW, England

Markus Pfeiffer
Email: markus.pfeiffer@morphism.de
Homepage: https://www.morphism.de/~markusp/

Christopher Russell

Jack Schmidt
Email: jack.schmidt@uky.edu
Homepage: https://www.ms.uky.edu/~jack/

Sergio Siccha
Email: sergio.siccha@gmail.com

Finn Smith
Email: fls3@st-andrews.ac.uk
Homepage: https://flsmith.github.io/
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Ben Spiers
Email: bspiers972@outlook.com

Nicolas Thiéry
Email: nthiery@users.sf.net
Homepage: https://nicolas.thiery.name/

Maria Tsalakou
Email: mt200@st-andrews.ac.uk
Homepage: https://mariatsalakou.github.io/
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Chris Wensley
Email: cdwensley.maths@btinternet.com

Murray Whyte
Email: mw231@st-andrews.ac.uk
Address:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland

Wilf A. Wilson
Email: gap@wilf-wilson.net
Homepage: https://wilf.me

Michael Young
Email: mct25@st-andrews.ac.uk
Homepage: https://mtorpey.github.io/
Address:
Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland

Fabian Zickgraf
Email: f.zickgraf@dashdos.com

Abstract

The Semigroups package is a GAP package for semigroups, and monoids. There are particularly efficient methods for finitely presented semigroups and monoids, and for semigroups and monoids consisting of transformations, partial permutations, bipartitions, partitioned binary relations, subsemigroups of regular Rees 0-matrix semigroups, and matrices of various semirings including boolean matrices, matrices over finite fields, and certain tropical matrices. Semigroups contains efficient methods for creating semigroups, monoids, and inverse semigroups and monoids, calculating their Green's structure, ideals, size, elements, group of units, small generating sets, testing membership, finding the inverses of a regular element, factorizing elements over the generators, and so on. It is possible to test if a semigroup satisfies a particular property, such as if it is regular, simple, inverse, completely regular, and a large number of further properties. There are methods for finding presentations for a semigroup, the congruences of a semigroup, the maximal subsemigroups of a finite semigroup, smaller degree partial permutation representations, and the character tables of inverse semigroups. There are functions for producing pictures of the Green's structure of a semigroup, and for drawing graphical representations of certain types of elements.

Copyright

© by J. D. Mitchell et al.

Semigroups is free software; you can redistribute it and/or modify it, under the terms of the GNU General Public License, version 3 of the License, or (at your option) any later, version.

Acknowledgements

The authors of the Semigroups package would like to thank:

Manuel Delgado

who contributed to the function DotString (16.1-1).

Casey Donoven and Rhiannon Dougall

for their contribution to the development of the algorithms for maximal subsemigroups and smaller degree partial permutation representations.

James East

who contributed to the part of the package relating to bipartitions. We also thank the University of Western Sydney for their support of the development of this part of the package.

Zak Mesyan

who contributed to the code for graph inverse semigroups; see Section 7.10.

Yann Péresse and Yanhui Wang

who contributed to the attribute MunnSemigroup (7.2-1).

Jhevon Smith and Ben Steinberg

who contributed the function CharacterTableOfInverseSemigroup (11.14-10).

We would also like to acknowledge the support of: EPSRC grant number GR/S/56085/01; the Carnegie Trust for the Universities of Scotland for funding the PhD scholarships of Julius Jonušas and Wilf A. Wilson when they worked on this project; the Engineering and Physical Sciences Research Council (EPSRC) for funding the PhD scholarships of F. Smith (EP/N509759/1) and M. Young (EP/M506631/1) when they worked on this project.

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