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# Semigroups

## A package for semigroups and monoids

5.2.0

1 December 2022

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/

Joe Edwards
Email: je53@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://www.st-andrews.ac.uk/computer-science/people/obk1/
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

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

### 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.

### Contents

1 The Semigroups package
1.1 Introduction
1.2 Overview
2 Installing Semigroups
2.1 For those in a hurry
2.2 Compiling the kernel module
2.3 Rebuilding the documentation
2.4 Testing your installation
2.4-1 SemigroupsTestInstall
2.4-2 SemigroupsTestStandard
2.4-3 SemigroupsTestExtreme
2.4-4 SemigroupsTestAll
2.5 More information during a computation
2.5-1 InfoSemigroups
3 Bipartitions and blocks
3.1 The family and categories of bipartitions
3.1-1 IsBipartition
3.1-2 IsBipartitionCollection
3.2 Creating bipartitions
3.2-1 Bipartition
3.2-2 BipartitionByIntRep
3.2-3 IdentityBipartition
3.2-4 LeftOne
3.2-5 RightOne
3.2-6 StarOp
3.2-7 RandomBipartition
3.3 Changing the representation of a bipartition
3.3-1 AsBipartition
3.3-2 AsBlockBijection
3.3-3 AsTransformation
3.3-4 AsPartialPerm
3.3-5 AsPermutation
3.4 Operators for bipartitions
3.4-1 PartialPermLeqBipartition
3.4-2 NaturalLeqPartialPermBipartition
3.4-3 NaturalLeqBlockBijection
3.4-4 PermLeftQuoBipartition
3.5 Attributes for bipartitons
3.5-1 DegreeOfBipartition
3.5-2 RankOfBipartition
3.5-3 ExtRepOfObj
3.5-4 IntRepOfBipartition
3.5-5 RightBlocks
3.5-6 LeftBlocks
3.5-7 NrLeftBlocks
3.5-8 NrRightBlocks
3.5-9 NrBlocks
3.5-10 DomainOfBipartition
3.5-11 CodomainOfBipartition
3.5-12 IsTransBipartition
3.5-13 IsDualTransBipartition
3.5-14 IsPermBipartition
3.5-15 IsPartialPermBipartition
3.5-16 IsBlockBijection
3.5-17 IsUniformBlockBijection
3.5-18 CanonicalBlocks
3.6 Creating blocks and their attributes
3.6-1 IsBlocks
3.6-2 BLOCKS_NC
3.6-3 ExtRepOfObj
3.6-4 RankOfBlocks
3.6-5 DegreeOfBlocks
3.6-6 ProjectionFromBlocks
3.7 Actions on blocks
3.7-1 OnRightBlocks
3.7-2 OnLeftBlocks
3.8 Semigroups of bipartitions
3.8-1 IsBipartitionSemigroup
3.8-2 IsBlockBijectionSemigroup
3.8-3 IsPartialPermBipartitionSemigroup
3.8-4 IsPermBipartitionGroup
3.8-5 DegreeOfBipartitionSemigroup
4 Partitioned binary relations (PBRs)
4.1 The family and categories of PBRs
4.1-1 IsPBR
4.1-2 IsPBRCollection
4.2 Creating PBRs
4.2-1 PBR
4.2-2 RandomPBR
4.2-3 EmptyPBR
4.2-4 IdentityPBR
4.2-5 UniversalPBR
4.3 Changing the representation of a PBR
4.3-1 AsPBR
4.3-2 AsTransformation
4.3-3 AsPartialPerm
4.3-4 AsPermutation
4.4 Operators for PBRs
4.5 Attributes for PBRs
4.5-1 StarOp
4.5-2 DegreeOfPBR
4.5-3 ExtRepOfObj
4.5-4 PBRNumber
4.5-5 IsEmptyPBR
4.5-6 IsIdentityPBR
4.5-7 IsUniversalPBR
4.5-8 IsBipartitionPBR
4.5-9 IsTransformationPBR
4.5-10 IsDualTransformationPBR
4.5-11 IsPartialPermPBR
4.5-12 IsPermPBR
4.6 Semigroups of PBRs
4.6-1 IsPBRSemigroup
4.6-2 DegreeOfPBRSemigroup
5 Matrices over semirings
5.1 Creating matrices over semirings
5.1-1 IsMatrixOverSemiring
5.1-2 IsMatrixOverSemiringCollection
5.1-3 DimensionOfMatrixOverSemiring
5.1-4 DimensionOfMatrixOverSemiringCollection
5.1-5 Matrix
5.1-6 AsMatrix
5.1-7 RandomMatrix
5.1-8 Matrix filters
5.1-9 Matrix collection filters
5.1-10 AsList
5.1-11 ThresholdTropicalMatrix
5.1-12 ThresholdNTPMatrix
5.2 Operators for matrices over semirings
5.3 Boolean matrices
5.3-1 BooleanMat
5.3-2 AsBooleanMat
`5.3-3 \in`
5.3-4 OnBlist
5.3-5 Successors
5.3-6 BooleanMatNumber
5.3-7 BlistNumber
5.3-8 CanonicalBooleanMat
5.3-9 IsRowTrimBooleanMat
5.3-10 IsSymmetricBooleanMat
5.3-11 IsReflexiveBooleanMat
5.3-12 IsTransitiveBooleanMat
5.3-13 IsAntiSymmetricBooleanMat
5.3-14 IsTotalBooleanMat
5.3-15 IsPartialOrderBooleanMat
5.3-16 IsEquivalenceBooleanMat
5.3-17 IsTransformationBooleanMat
5.4 Matrices over finite fields
5.4-1 RowSpaceBasis
5.4-2 RightInverse
5.5 Matrices over the integers
5.5-1 InverseOp
5.5-2 IsTorsion
5.5-3 Order
5.6 Max-plus and min-plus matrices
5.6-1 InverseOp
5.6-2 RadialEigenvector
5.6-3 SpectralRadius
5.6-4 UnweightedPrecedenceDigraph
5.7 Matrix semigroups
5.7-1 Matrix semigroup filters
5.7-2 Matrix monoid filters
5.7-3 IsFinite
5.7-4 IsTorsion
5.7-5 NormalizeSemigroup
6 Semigroups and monoids defined by generating sets
6.1 Underlying algorithms
6.1-1 Acting semigroups
6.1-2 IsActingSemigroup
6.1-3 The Froidure-Pin Algorithm
6.1-4 CanUseFroidurePin
6.2 Semigroups represented by generators
6.2-1 InverseMonoidByGenerators
6.3 Options when creating semigroups
6.3-1 SEMIGROUPS.DefaultOptionsRec
6.4 Subsemigroups and supersemigroups
6.4-1 ClosureSemigroup
6.4-2 SubsemigroupByProperty
6.4-3 InverseSubsemigroupByProperty
6.5 Changing the representation of a semigroup
6.5-1 IsomorphismSemigroup
6.5-2 IsomorphismMonoid
6.5-3 AsSemigroup
6.5-4 AsMonoid
6.5-5 IsomorphismPermGroup
6.5-6 RZMSNormalization
6.5-7 RMSNormalization
6.5-8 IsomorphismReesMatrixSemigroup
6.5-9 AntiIsomorphismDualFpSemigroup
6.6 Random semigroups
6.6-1 RandomSemigroup
7 Standard examples
7.1 Transformation semigroups
7.1-1 CatalanMonoid
7.1-2 EndomorphismsPartition
7.1-3 PartialTransformationMonoid
7.1-4 SingularTransformationSemigroup
7.1-5 Semigroups of order-preserving transformations
7.1-6 EndomorphismMonoid
7.2 Semigroups of partial permutations
7.2-1 MunnSemigroup
7.2-2 RookMonoid
7.2-3 Inverse monoids of order-preserving partial permutations
7.3 Semigroups of bipartitions
7.3-1 PartitionMonoid
7.3-2 BrauerMonoid
7.3-3 JonesMonoid
7.3-4 PartialJonesMonoid
7.3-5 AnnularJonesMonoid
7.3-6 MotzkinMonoid
7.3-7 DualSymmetricInverseSemigroup
7.3-8 UniformBlockBijectionMonoid
7.3-9 PlanarPartitionMonoid
7.3-10 ModularPartitionMonoid
7.3-11 ApsisMonoid
7.4 Standard PBR semigroups
7.4-1 FullPBRMonoid
7.5 Semigroups of matrices over a finite field
7.5-1 FullMatrixMonoid
7.5-2 SpecialLinearMonoid
7.5-3 IsFullMatrixMonoid
7.6 Semigroups of boolean matrices
7.6-1 FullBooleanMatMonoid
7.6-2 RegularBooleanMatMonoid
7.6-3 ReflexiveBooleanMatMonoid
7.6-4 HallMonoid
7.6-5 GossipMonoid
7.6-6 TriangularBooleanMatMonoid
7.7 Semigroups of matrices over a semiring
7.7-1 FullTropicalMaxPlusMonoid
7.7-2 FullTropicalMinPlusMonoid
7.8 Examples in various representations
7.8-1 TrivialSemigroup
7.8-2 MonogenicSemigroup
7.8-3 RectangularBand
7.8-4 ZeroSemigroup
7.8-5 LeftZeroSemigroup
7.8-6 BrandtSemigroup
7.9 Free bands
7.9-1 FreeBand
7.9-2 IsFreeBandCategory
7.9-3 IsFreeBand
7.9-4 IsFreeBandElement
7.9-5 IsFreeBandElementCollection
7.9-6 IsFreeBandSubsemigroup
7.9-7 ContentOfFreeBandElement
7.9-8 EqualInFreeBand
7.9-9 GreensDClassOfElement
7.9-10 Operators
7.10 Graph inverse semigroups
7.10-1 GraphInverseSemigroup
7.10-2 Range
7.10-3 IsVertex
7.10-4 IsGraphInverseSemigroup
7.10-5 GraphOfGraphInverseSemigroup
7.10-6 IsGraphInverseSemigroupElementCollection
7.10-7 IsGraphInverseSubsemigroup
7.10-8 VerticesOfGraphInverseSemigroup
7.10-9 IndexOfVertexOfGraphInverseSemigroup
7.11 Free inverse semigroups
7.11-1 FreeInverseSemigroup
7.11-2 IsFreeInverseSemigroupCategory
7.11-3 IsFreeInverseSemigroup
7.11-4 IsFreeInverseSemigroupElement
7.11-5 IsFreeInverseSemigroupElementCollection
7.11-6 CanonicalForm
7.11-7 MinimalWord
7.11-8 Displaying free inverse semigroup elements
7.11-9 Operators for free inverse semigroup elements
8 Standard constructions
8.1 Products of semigroups
8.1-1 DirectProduct
8.1-2 WreathProduct
8.2 Dual semigroups
8.2-1 DualSemigroup
8.2-2 IsDualSemigroupRep
8.2-3 IsDualSemigroupElement
8.2-4 AntiIsomorphismDualSemigroup
8.3 Strong semilattices of semigroups
8.3-1 StrongSemilatticeOfSemigroups
8.3-2 SSSE
8.3-3 IsSSSE
8.3-4 IsStrongSemilatticeOfSemigroups
8.3-5 SemilatticeOfStrongSemilatticeOfSemigroups
8.3-6 SemigroupsOfStrongSemilatticeOfSemigroups
8.3-7 HomomorphismsOfStrongSemilatticeOfSemigroups
8.4 McAlister triple semigroups
8.4-1 IsMcAlisterTripleSemigroup
8.4-2 McAlisterTripleSemigroup
8.4-3 McAlisterTripleSemigroupGroup
8.4-4 McAlisterTripleSemigroupPartialOrder
8.4-5 McAlisterTripleSemigroupSemilattice
8.4-6 McAlisterTripleSemigroupAction
8.4-7 IsMcAlisterTripleSemigroupElement
8.4-8 McAlisterTripleSemigroupElement
9 Ideals
9.1 Creating ideals
9.1-1 SemigroupIdeal
9.1-2 Ideals
9.2 Attributes of ideals
9.2-1 GeneratorsOfSemigroupIdeal
9.2-2 MinimalIdealGeneratingSet
9.2-3 SupersemigroupOfIdeal
10 Green's relations
10.1 Creating Green's classes and representatives
10.1-1 XClassOfYClass
10.1-2 GreensXClassOfElement
10.1-3 GreensXClassOfElementNC
10.1-4 GreensXClasses
10.1-5 XClassReps
10.1-6 MinimalDClass
10.1-7 MaximalXClasses
10.1-8 NrRegularDClasses
10.1-9 NrXClasses
10.1-10 PartialOrderOfXClasses
10.1-11 LengthOfLongestDClassChain
10.1-12 IsGreensDGreaterThanFunc
10.2 Iterators and enumerators of classes and representatives
10.2-1 IteratorOfXClassReps
10.2-2 IteratorOfXClasses
10.3 Properties of Green's classes
10.3-1 Less than for Green's classes
10.3-2 IsRegularGreensClass
10.3-3 IsGreensClassNC
10.4 Attributes of Green's classes
10.4-1 GroupHClass
10.4-2 SchutzenbergerGroup
10.4-3 StructureDescriptionSchutzenbergerGroups
10.4-4 StructureDescriptionMaximalSubgroups
10.4-5 MultiplicativeNeutralElement
10.4-6 StructureDescription
10.4-7 InjectionPrincipalFactor
10.4-8 PrincipalFactor
11 Attributes and operations for semigroups
11.1 Accessing the elements of a semigroup
11.1-1 AsListCanonical
11.1-2 PositionCanonical
11.1-3 Enumerate
11.1-4 IsEnumerated
11.2 Cayley graphs
11.2-1 RightCayleyDigraph
11.3 Random elements of a semigroup
11.3-1 Random
11.4 Properties of elements in a semigroup
11.4-1 IndexPeriodOfSemigroupElement
11.4-2 SmallestIdempotentPower
11.5 Expressing semigroup elements as words in generators
11.5-1 EvaluateWord
11.5-2 Factorization
11.5-3 MinimalFactorization
11.5-4 NonTrivialFactorization
11.6 Generating sets
11.6-1 Generators
11.6-2 SmallGeneratingSet
11.6-3 IrredundantGeneratingSubset
11.6-4 MinimalSemigroupGeneratingSet
11.6-5 GeneratorsSmallest
11.6-6 IndecomposableElements
11.7 Minimal ideals and multiplicative zeros
11.7-1 MinimalIdeal
11.7-2 RepresentativeOfMinimalIdeal
11.7-3 MultiplicativeZero
11.7-4 UnderlyingSemigroupOfSemigroupWithAdjoinedZero
11.8 Group of units and identity elements
11.8-1 GroupOfUnits
11.9 Idempotents
11.9-1 Idempotents
11.9-2 NrIdempotents
11.9-3 IdempotentGeneratedSubsemigroup
11.10 Maximal subsemigroups
11.10-1 MaximalSubsemigroups
11.10-2 NrMaximalSubsemigroups
11.10-3 IsMaximalSubsemigroup
11.11 Attributes of transformations and transformation semigroups
11.11-1 ComponentRepsOfTransformationSemigroup
11.11-2 ComponentsOfTransformationSemigroup
11.11-3 CyclesOfTransformationSemigroup
11.11-4 DigraphOfAction
11.11-5 DigraphOfActionOnPoints
11.11-6 FixedPointsOfTransformationSemigroup
11.11-7 IsTransitive
11.11-8 SmallestElementSemigroup
11.11-9 CanonicalTransformation
11.11-10 IsConnectedTransformationSemigroup
11.12 Attributes of partial perm semigroups
11.12-1 ComponentRepsOfPartialPermSemigroup
11.12-2 ComponentsOfPartialPermSemigroup
11.12-3 CyclesOfPartialPerm
11.12-4 CyclesOfPartialPermSemigroup
11.13 Attributes of Rees (0-)matrix semigroups
11.13-1 RZMSDigraph
11.13-2 RZMSConnectedComponents
11.14 Attributes of inverse semigroups
11.14-1 NaturalLeqInverseSemigroup
11.14-2 JoinIrreducibleDClasses
11.14-3 MajorantClosure
11.14-4 Minorants
11.14-5 PrimitiveIdempotents
11.14-6 RightCosetsOfInverseSemigroup
11.14-7 SameMinorantsSubgroup
11.14-8 SmallerDegreePartialPermRepresentation
11.14-9 VagnerPrestonRepresentation
11.14-10 CharacterTableOfInverseSemigroup
11.14-11 EUnitaryInverseCover
11.15 Nambooripad partial order
11.15-1 NambooripadLeqRegularSemigroup
11.15-2 NambooripadPartialOrder
12 Properties of semigroups
12.1 Arbitrary semigroups
12.1-1 IsBand
12.1-2 IsBlockGroup
12.1-3 IsCommutativeSemigroup
12.1-4 IsCompletelyRegularSemigroup
12.1-5 IsCongruenceFreeSemigroup
12.1-6 IsSurjectiveSemigroup
12.1-7 IsGroupAsSemigroup
12.1-8 IsIdempotentGenerated
12.1-9 IsLeftSimple
12.1-10 IsLeftZeroSemigroup
12.1-11 IsMonogenicSemigroup
12.1-12 IsMonogenicMonoid
12.1-13 IsMonoidAsSemigroup
12.1-14 IsOrthodoxSemigroup
12.1-15 IsRectangularBand
12.1-16 IsRectangularGroup
12.1-17 IsRegularSemigroup
12.1-18 IsRightZeroSemigroup
12.1-19 IsXTrivial
12.1-20 IsSemigroupWithAdjoinedZero
12.1-21 IsSemilattice
12.1-22 IsSimpleSemigroup
12.1-23 IsSynchronizingSemigroup
12.1-24 IsUnitRegularMonoid
12.1-25 IsZeroGroup
12.1-26 IsZeroRectangularBand
12.1-27 IsZeroSemigroup
12.1-28 IsZeroSimpleSemigroup
12.1-29 IsSelfDualSemigroup
12.2 Inverse semigroups
12.2-1 IsCliffordSemigroup
12.2-2 IsBrandtSemigroup
12.2-3 IsEUnitaryInverseSemigroup
12.2-4 IsFInverseSemigroup
12.2-5 IsFInverseMonoid
12.2-6 IsFactorisableInverseMonoid
12.2-7 IsJoinIrreducible
12.2-8 IsMajorantlyClosed
12.2-9 IsMonogenicInverseSemigroup
12.2-10 IsMonogenicInverseMonoid
13 Congruences
13.1 Semigroup congruence objects
13.1-1 IsSemigroupCongruence
13.1-2 IsLeftSemigroupCongruence
13.1-3 IsRightSemigroupCongruence
13.2 Creating congruences
13.2-1 SemigroupCongruence
13.2-2 LeftSemigroupCongruence
13.2-3 RightSemigroupCongruence
13.3 Congruence classes
13.3-1 IsCongruenceClass
13.3-2 IsLeftCongruenceClass
13.3-3 IsRightCongruenceClass
13.3-4 NonTrivialEquivalenceClasses
13.3-5 EquivalenceRelationLookup
13.3-6 EquivalenceRelationCanonicalLookup
13.3-7 EquivalenceRelationCanonicalPartition
13.3-8 OnLeftCongruenceClasses
13.3-9 OnRightCongruenceClasses
13.4 Finding the congruences of a semigroup
13.4-1 CongruencesOfSemigroup
13.4-2 MinimalCongruencesOfSemigroup
13.4-3 PrincipalCongruencesOfSemigroup
13.4-4 IsCongruencePoset
13.4-5 LatticeOfCongruences
13.4-6 PosetOfPrincipalCongruences
13.4-7 CongruencesOfPoset
13.4-8 UnderlyingSemigroupOfCongruencePoset
13.4-9 PosetOfCongruences
13.4-10 JoinSemilatticeOfCongruences
13.4-11 MinimalCongruences
13.4-12 NumberOfRightCongruences
13.4-13 IteratorOfRightCongruences
13.5 Comparing congruences
13.5-1 IsSubrelation
13.5-2 IsSuperrelation
13.5-3 MeetSemigroupCongruences
13.5-4 JoinSemigroupCongruences
13.6 Congruences on Rees matrix semigroups
13.6-1 IsRMSCongruenceByLinkedTriple
13.6-2 RMSCongruenceByLinkedTriple
13.6-3 IsRMSCongruenceClassByLinkedTriple
13.6-4 RMSCongruenceClassByLinkedTriple
13.6-5 IsLinkedTriple
13.6-6 AsSemigroupCongruenceByGeneratingPairs
13.7 Congruences on inverse semigroups
13.7-1 IsInverseSemigroupCongruenceByKernelTrace
13.7-2 InverseSemigroupCongruenceByKernelTrace
13.7-3 AsInverseSemigroupCongruenceByKernelTrace
13.7-4 KernelOfSemigroupCongruence
13.7-5 TraceOfSemigroupCongruence
13.7-6 IsInverseSemigroupCongruenceClassByKernelTrace
13.7-7 MinimumGroupCongruence
13.8 Congruences on graph inverse semigroups
13.8-1 IsCongruenceByWangPair
13.8-2 CongruenceByWangPair
13.8-3 AsCongruenceByWangPair
13.8-4 GeneratingCongruencesOfLattice
13.9 Rees congruences
13.9-1 SemigroupIdealOfReesCongruence
13.9-2 IsReesCongruenceClass
13.10 Universal congruences
13.10-1 IsUniversalSemigroupCongruence
13.10-2 IsUniversalSemigroupCongruenceClass
13.10-3 UniversalSemigroupCongruence
14 Semigroup homomorphisms
14.1 Homomorphisms of arbitrary semigroups
14.1-1 SemigroupHomomorphismByImages
14.1-2 SemigroupHomomorphismByFunctionNC
14.1-3 IsSemigroupHomomorphismByImages
14.1-4 IsSemigroupHomomorphismByFunction
14.1-5 AsSemigroupHomomorphismByImages
14.1-6 AsSemigroupHomomorphismByFunction
14.1-7 KernelOfSemigroupHomomorphism
14.2 Isomorphisms of arbitrary semigroups
14.2-1 IsIsomorphicSemigroup
14.2-2 SmallestMultiplicationTable
14.2-3 CanonicalMultiplicationTable
14.2-4 CanonicalMultiplicationTablePerm
14.2-5 OnMultiplicationTable
14.2-6 IsomorphismSemigroups
14.2-7 AutomorphismGroup
14.2-8 SemigroupIsomorphismByImages
14.2-9 SemigroupIsomorphismByFunctionNC
14.2-10 IsSemigroupIsomorphismByFunction
14.2-11 AsSemigroupIsomorphismByFunction
14.2-12 SmallerDegreeTransformationRepresentation
14.2-13 MinimalFaithfulTransformationDegree
14.3 Isomorphisms of Rees (0-)matrix semigroups
14.3-1 IsRMSIsoByTriple
14.3-2 RMSIsoByTriple
14.3-3 ELM_LIST
14.3-4 CompositionMapping2
14.3-5 ImagesElm
14.3-6 CanonicalReesZeroMatrixSemigroup
14.3-7 Operators for isomorphisms of Rees (0-)matrix semigroups
15 Finitely presented semigroups and Tietze transformations
15.1 Changing representation for words and strings
15.1-1 WordToString
15.1-2 RandomWord
15.1-3 StandardiseWord
15.1-4 StringToWord
15.2 Helper functions
15.2-1 ParseRelations
15.2-2 ElementOfFpSemigroup
15.2-3 ElementOfFpMonoid
15.2-4 FreeMonoidAndAssignGeneratorVars
15.3 Creating Tietze transformation objects
15.3-1 StzPresentation
15.3-2 IsStzPresentation
15.3-3 GeneratorsOfStzPresentation
15.3-4 RelationsOfStzPresentation
15.3-5 UnreducedFpSemigroup
15.3-6 Length
15.4 Printing Tietze transformation objects
15.4-1 StzPrintRelations
15.4-2 StzPrintRelation
15.4-3 StzPrintGenerators
15.4-4 StzPrintPresentation
15.5 Changing Tietze transformation objects
15.5-1 StzAddRelation
15.5-2 StzRemoveRelation
15.5-3 StzAddGenerator
15.5-4 StzRemoveGenerator
15.5-5 StzSubstituteRelation
15.6 Converting a Tietze transformation object into a fp semigroup
15.6-1 TietzeForwardMap
15.6-2 TietzeBackwardMap
15.6-3 StzIsomorphism
15.7 Automatically simplifying a Tietze transformation object
15.7-1 StzSimplifyOnce
15.7-2 StzSimplifyPresentation
15.8 Automatically simplifying an fp semigroup
15.8-1 SimplifyFpSemigroup
15.8-2 SimplifiedFpSemigroup
15.8-3 UnreducedFpSemigroup
15.8-4 FpTietzeIsomorphism
16 Visualising semigroups and elements
16.1 `dot` pictures
16.1-1 DotString
16.1-2 DotString
16.1-3 DotSemilatticeOfIdempotents
16.1-4 DotLeftCayleyDigraph
16.2 `tex` output
16.2-1 TexString
16.3 `tikz` pictures
16.3-1 TikzString
16.3-2 TikzLeftCayleyDigraph
17 IO
17.1 Reading and writing elements to a file
17.1-1 ReadGenerators
17.1-2 WriteGenerators
17.1-3 IteratorFromGeneratorsFile
17.2 Reading and writing multiplication tables to a file
17.2-1 ReadMultiplicationTable
17.2-2 WriteMultiplicationTable
17.2-3 IteratorFromMultiplicationTableFile
18 Translations
18.1 Methods for translations
18.1-1 IsXTranslation
18.1-2 IsBitranslation
18.1-3 IsXTranslationCollection
18.1-4 XPartOfBitranslation
18.1-5 XTranslation
18.1-6 Bitranslation
18.1-7 UnderlyingSemigroup
18.1-8 XTranslationsSemigroupOfFamily
18.1-9 TypeXTranslationSemigroupElements
18.1-10 XTranslations
18.1-11 TranslationalHull
18.1-12 NrXTranslations
18.1-13 InnerXTranslations
18.1-14 InnerTranslationalHull
18.1-15 UnderlyingRepresentatives
18.1-16 ImageSetOfTranslation
References
Index
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