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Semigroups

Version 2.8.0

J. D. Mitchell
Email: jdm3@st-and.ac.uk
Homepage: http://tinyurl.com/jdmitchell

Manuel Delgado

James East

Attila Egri-Nagy

Nicholas Ham

Julius Jonušas

Markus Pfeiffer

Ben Steinberg

Jhevon Smith

Michael Torpey

Wilf Wilson

Abstract

The Semigroups package is a GAP package containing methods for semigroups, monoids, and inverse semigroups, principally of transformations, partial permutations, bipartitions, subsemigroups of regular Rees 0-matrix semigroups, free inverse semigroups, free bands, and semigroups of matrices over finite fields.

Semigroups contains more efficient methods than those available in the GAP library (and in many cases more efficient than any other software) for creating semigroups, monoids, and inverse semigroup, 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 many more. It is also possible to test if a semigroup satisfies a particular property, such as if it is regular, simple, inverse, completely regular, and a variety of further properties.

There are methods for finding congruences of certain types of semigroups, the normalizer of a semigroup in a permutation group, the maximal subsemigroups of a finite semigroup, and 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 bipartitions.

Copyright

© 2011-16 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 as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.

Acknowledgements

I would like to thank P. von Bunau, A. Distler, S. Linton, C. Nehaniv, J. Neubueser, M. R. Quick, E. F. Robertson, and N. Ruskuc for their help and suggestions. Special thanks go to J. Araujo for his mathematical suggestions and to M. Neunhoeffer for his invaluable help in improving the efficiency of the package.

Manuel Delgado and Attila Egri-Nagy contributed to the functions Splash (4.8-1) and DotDClasses (4.8-2).

James East, Attila Egri-Nagy, and Markus Pfeiffer contributed to the part of the package relating to bipartitions. I would like to thank the University of Western Sydney for their support of the development of this part of the package.

Nick Ham contributed many of the standard examples of bipartition semigroups.

Julius Jonušas contributed the part of the package relating to free inverse semigroups, and contributed to the code for ideals.

Yann Peresse and Yanhui Wang contributed to the function MunnSemigroup (2.5-13).

Jhevon Smith and Ben Steinberg contributed the function CharacterTableOfInverseSemigroup (4.7-16).

Michael Torpey contributed the part of the package relating to congruences of Rees (0-)matrix semigroups.

Wilf Wilson contributed to the part of the package relating maximal subsemigroups and smaller degree partial permutation representations of inverse semigroups. We are also grateful to C. Donoven and R. Hancock for their contribution to the development of the algorithms for maximal subsemigroups and smaller degree partial permutation representations.

Markus Pfeiffer contributed the majority of the code relating to semigroups of matrices over finite fields.

We would also like to acknowledge the support of the Centre of Algebra at the University of Lisbon, and of EPSRC grant number GR/S/56085/01.

Contents

1 The Semigroups package
2 Creating semigroups and monoids
3 Ideals
4 Determining the structure of a semigroup
5 Bipartitions and blocks
6 Free inverse semigroups and free bands
7 Matrix semigroups
8 Congruences
9 Homomorphisms
10 Orbits
References
Index

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