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# Smallsemi

## A library of small semigroups

Version 0.6.13

28 February 2022

Andreas Distler
Email: a.distler@tu-bs.de

James Mitchell
Email: jdm3@st-and.ac.uk
Homepage: http://tinyurl.com/jdmitchell
Address:
Mathematical Institute
North Haugh
St Andrews
Fife
KY16 9SS
Scotland

### Copyright

© 2008-19 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 'LICENSE'; for the latest version see https://www.gnu.org/licenses/#GPL.

This file is part of Smallsemi, though as documentation it is released under the GNU Free Documentation License (see https://www.gnu.org/licenses/#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 find any bugs or have any suggestions or comments, we would very much appreciate it if you would let us know via our issue tracker (see https://github.com/gap-packages/smallsemi/issues).

### 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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