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# Smallsemi

## A GAP package

Version 0.6.8

Andreas Distler
Email: a.distler@tu-bs.de
Address:
Institut Computational Mathematics
Rebenring 31 (A14)
38106 Braunschweig
Germany

J. D. Mitchell
Email: jdm3@st-and.ac.uk
Homepage: http://www-groups.mcs.st-and.ac.uk/~jamesm
Address:
Mathematical Institute
North Haugh
St Andrews, Fife
KY16 9SS
Scotland, UK

### Copyright

© 2008-14 A. Distler & J. D. Mitchell.

Smallsemi 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.

Smallsemi is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

A copy of the GNU General Public License is available in the file 'GPLv3'; for the latest version see`http://www.gnu.org/licenses/.

This file is part of Smallsemi, though as documentation it is released under the GNU Free Documentation License (see http://www.gnu.org/licenses/licenses.html#FDL).

### Acknowledgements

We would like to thank Tom Kelsey for making this library possible by running all the initial computations in Minion [GJM06].
The first author acknowledges financial support of the University of St Andrews. The second author acknowledges support of EPSRC grant number GR/S/56085/01.

### Colophon

If you use Smallsemi, please tell us by sending an email to a.distler@tu-bs.de or jdm3@st-and.ac.uk.

If you find any bugs or have any suggestions or comments, we would very much appreciate it if you would let us know. Also, we would like to hear about applications of this software.

### Contents

1 Introduction
1.1 Requirements
1.1-1 Operating System
1.1-2 RAM
1.1-3 Disk Space
1.2 Installation and Setup
1.2-1 Download and Extract Smallsemi
1.2-2 Contents
1.2-3 Loading
1.2-4 Memory Issues
1.2-5 Testing
1.2-6 Customizing
2 The Data in the Library
2.1 Creation of the Semigroups
2.2 Storing the Semigroups
3 Extended Examples
3.1 Lists, Enumerators and Iterators of Semigroups
3.1-1 Precomputed properties
3.1-2 User functions
3.1-3 Semigroups of order 8
3.2 Identifying Semigroups
4 Functionality
4.1 Individual Semigroups
4.1-1 SmallSemigroup
4.1-2 IsSmallSemigroup
4.1-3 IsSmallSemigroupElt
4.1-4 RecoverMultiplicationTable
4.1-5 SemigroupByMultiplicationTableNC
4.1-6 IdSmallSemigroup
4.1-7 EquivalenceSmallSemigroup
4.1-8 InfoSmallsemi
4.1-9 UnloadSmallsemiData
4.2 Properties of Semigroups
4.2-1 Annihilators
4.2-2 DiagonalOfMultiplicationTable
4.2-3 DisplaySmallSemigroup
4.2-4 IndexPeriod
4.2-5 IsBand
4.2-6 IsBrandtSemigroup
4.2-7 IsCliffordSemigroup
4.2-8 IsCommutativeSemigroup
4.2-9 IsCompletelyRegularSemigroup
4.2-10 IsFullTransformationSemigroupCopy
4.2-11 IsGroupAsSemigroup
4.2-12 IsIdempotentGenerated
4.2-13 IsInverseSemigroup
4.2-14 IsLeftZeroSemigroup
4.2-15 IsMonogenicSemigroup
4.2-16 IsMonoidAsSemigroup
4.2-17 IsMultSemigroupOfNearRing
4.2-18 IsNGeneratedSemigroup
4.2-19 IsNIdempotentSemigroup
4.2-20 IsNilpotentSemigroup
4.2-21 IsOrthodoxSemigroup
4.2-22 IsRectangularBand
4.2-23 IsRegularSemigroup
4.2-24 IsRightZeroSemigroup
4.2-25 IsSelfDualSemigroup
4.2-26 IsSemigroupWithClosedIdempotents
4.2-27 IsSemigroupWithZero
4.2-28 IsSimpleSemigroup
4.2-29 IsSingularSemigroupCopy
4.2-30 IsZeroGroup
4.2-31 IsZeroSemigroup
4.2-32 IsZeroSimpleSemigroup
4.2-33 MinimalGeneratingSet
4.2-34 NilpotencyDegree
4.3 Nilpotent semigroups by coclass
4.3-1 NilpotentSemigroupsByCoclass
4.4 Starred Green's relations
4.4-1 IsStarRelation
4.4-2 RStarRelation
4.4-3 RStarClass
4.4-4 RStarClass
4.4-5 IsStarClass
4.4-6 RStarClasses
4.5 Families of Semigroups
4.5-1 AllSmallSemigroups
4.5-2 EnumeratorOfSmallSemigroups
4.5-3 EnumeratorOfSmallSemigroupsByIds
4.5-4 EnumeratorSortedOfSmallSemigroups
4.5-5 FuncsOfSmallSemisInEnum
4.5-6 FuncsOfSmallSemisInIter
4.5-7 IdsOfSmallSemigroups
4.5-8 IsEnumeratorOfSmallSemigroups
4.5-9 IsIdSmallSemigroup
4.5-10 IsIteratorOfSmallSemigroups
4.5-11 IteratorOfSmallSemigroups
4.5-12 NamesFuncsSmallSemisInEnum
4.5-13 NamesFuncsSmallSemisInIter
4.5-14 Nr3NilpotentSemigroups
4.5-15 NrSmallSemigroups
4.5-16 OneSmallSemigroup
4.5-17 PositionsOfSmallSemigroups
4.5-18 PositionsOfSmallSemisInEnum
4.5-19 PrecomputedSmallSemisInfo
4.5-20 RandomSmallSemigroup
4.5-21 SizesOfSmallSemisInEnum
4.5-22 SizesOfSmallSemisInIter
4.5-23 UpToIsomorphism
References
Index
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