  
  
                                  [1X Semigroups [101X
  
  
                     [1X A package for semigroups and monoids [101X
  
  
                                     5.5.4
  
  
                                 29 August 2025
  
  
                                 James Mitchell
  
                        Marina Anagnostopoulou-Merkouri
  
                                 Thomas Breuer
  
                                 Stuart Burrell
  
                                 Reinis Cirpons
  
                                Tom Conti-Leslie
  
                                 Joseph Edwards
  
                                Attila Egri-Nagy
  
                                  Luke Elliott
  
                             Fernando Flores Brito
  
                               Tillman Froehlich
  
                                    Nick Ham
  
                                 Robert Hancock
  
                                    Max Horn
  
                             Christopher Jefferson
  
                                 Julius Jonusas
  
                                Chinmaya Nagpal
  
                               Olexandr Konovalov
  
                             Artemis Konstantinidi
  
                                 Hyeokjun Kwon
  
                               Dima V. Pasechnik
  
                                Markus Pfeiffer
  
                              Christopher Russell
  
                                  Jack Schmidt
  
                                 Sergio Siccha
  
                                   Finn Smith
  
                                   Ben Spiers
  
                                 Nicolas Thiéry
  
                                 Maria Tsalakou
  
                                 Chris Wensley
  
                                  Murray Whyte
  
                                 Wilf A. Wilson
  
                                  Tianrun Yang
  
                                 Michael Young
  
                                Fabian Zickgraf
  
  
  
  James Mitchell
      Email:    [7Xmailto:jdm3@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://jdbm.me[107X
      Address:  [33X[0;14YMathematical  Institute,  North  Haugh, St Andrews, Fife, KY16
                9SS, Scotland[133X
  
  
  Marina Anagnostopoulou-Merkouri
      Email:    [7Xmailto:marina.anagnostopoulou-merkouri@bristol.ac.uk[107X
      Homepage: [7Xhttps://marinaanagno.github.io[107X
  Thomas Breuer
      Email:    [7Xmailto:sam@math.rwth-aachen.de[107X
      Homepage: [7Xhttps://www.math.rwth-aachen.de/~Thomas.Breuer/[107X
  Stuart Burrell
      Email:    [7Xmailto:stuartburrell1994@gmail.com[107X
      Homepage: [7Xhttps://stuartburrell.github.io[107X
  Reinis Cirpons
      Email:    [7Xmailto:rc234@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://reinisc.id.lv/[107X
      Address:  [33X[0;14YMathematical  Institute,  North  Haugh, St Andrews, Fife, KY16
                9SS, Scotland[133X
  
  
  Tom Conti-Leslie
      Email:    [7Xmailto:tom.contileslie@gmail.com[107X
      Homepage: [7Xhttps://tomcontileslie.com/[107X
  Joseph Edwards
      Email:    [7Xmailto:jde1@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://github.com/Joseph-Edwards[107X
      Address:  [33X[0;14YMathematical  Institute,  North  Haugh, St Andrews, Fife, KY16
                9SS, Scotland[133X
  
  
  Attila Egri-Nagy
      Email:    [7Xmailto:attila@egri-nagy.hu[107X
      Homepage: [7Xhttp://www.egri-nagy.hu[107X
  Luke Elliott
      Email:    [7Xmailto:le27@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://le27.github.io/Luke-Elliott/[107X
      Address:  [33X[0;14YMathematical  Institute,  North  Haugh, St Andrews, Fife, KY16
                9SS, Scotland[133X
  
  
  Fernando Flores Brito
      Email:    [7Xmailto:ffloresbrito@gmail.com[107X
  Tillman Froehlich
      Email:    [7Xmailto:trf1@st-andrews.ac.uk[107X
  Nick Ham
      Email:    [7Xmailto:nicholas.charles.ham@gmail.com[107X
      Homepage: [7Xhttps://n-ham.github.io[107X
  Robert Hancock
      Email:    [7Xmailto:robert.hancock@maths.ox.ac.uk[107X
      Homepage: [7Xhttps://sites.google.com/view/robert-hancock/[107X
  Max Horn
      Email:    [7Xmailto:mhorn@rptu.de[107X
      Homepage: [7Xhttps://www.quendi.de/math[107X
      Address:  [33X[0;14YFachbereich     Mathematik,     RPTU    Kaiserslautern-Landau,
                Gottlieb-Daimler-Straße 48, 67663 Kaiserslautern, Germany[133X
  
  
  Christopher Jefferson
      Email:    [7Xmailto:caj21@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://heather.cafe/[107X
      Address:  [33X[0;14YJack  Cole  Building, North Haugh, St Andrews, Fife, KY16 9SX,
                Scotland[133X
  
  
  Julius Jonusas
      Email:    [7Xmailto:j.jonusas@gmail.com[107X
      Homepage: [7Xhttp://julius.jonusas.work[107X
  Olexandr Konovalov
      Email:    [7Xmailto:obk1@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://olexandr-konovalov.github.io/[107X
      Address:  [33X[0;14YJack  Cole  Building, North Haugh, St Andrews, Fife, KY16 9SX,
                Scotland[133X
  
  
  Dima V. Pasechnik
      Email:    [7Xmailto:dmitrii.pasechnik@cs.ox.ac.uk[107X
      Homepage: [7Xhttp://users.ox.ac.uk/~coml0531/[107X
      Address:  [33X[0;14YPembroke College, St. Aldates, Oxford OX1 1DW, England[133X
  
  
  Markus Pfeiffer
      Email:    [7Xmailto:markus.pfeiffer@morphism.de[107X
      Homepage: [7Xhttps://markusp.morphism.de/[107X
  Jack Schmidt
      Email:    [7Xmailto:jack.schmidt@uky.edu[107X
      Homepage: [7Xhttps://www.ms.uky.edu/~jack/[107X
  Sergio Siccha
      Email:    [7Xmailto:sergio.siccha@gmail.com[107X
  Finn Smith
      Email:    [7Xmailto:fls3@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://flsmith.github.io/[107X
      Address:  [33X[0;14YMathematical  Institute,  North  Haugh, St Andrews, Fife, KY16
                9SS, Scotland[133X
  
  
  Nicolas Thiéry
      Email:    [7Xmailto:nthiery@users.sf.net[107X
      Homepage: [7Xhttps://nicolas.thiery.name/[107X
  Maria Tsalakou
      Email:    [7Xmailto:mt200@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://mariatsalakou.github.io/[107X
      Address:  [33X[0;14YMathematical  Institute,  North  Haugh, St Andrews, Fife, KY16
                9SS, Scotland[133X
  
  
  Chris Wensley
      Email:    [7Xmailto:cdwensley.maths@btinternet.com[107X
  Murray Whyte
      Email:    [7Xmailto:mw231@st-andrews.ac.uk[107X
      Address:  [33X[0;14YMathematical  Institute,  North  Haugh, St Andrews, Fife, KY16
                9SS, Scotland[133X
  
  
  Wilf A. Wilson
      Email:    [7Xmailto:gap@wilf-wilson.net[107X
      Homepage: [7Xhttps://wilf.me[107X
  Michael Young
      Email:    [7Xmailto:mct25@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://mtorpey.github.io/[107X
      Address:  [33X[0;14YJack  Cole  Building, North Haugh, St Andrews, Fife, KY16 9SX,
                Scotland[133X
  
  
  Fabian Zickgraf
      Email:    [7Xmailto:f.zickgraf@dashdos.com[107X
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  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.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© by J. D. Mitchell et al.[133X
  
  [33X[0;0Y[5XSemigroups[105X 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.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThe authors of the [5XSemigroups[105X package would like to thank:[133X
  
  [8X Manuel Delgado [108X
        [33X[0;6Ywho contributed to the function [2XDotString[102X ([14X16.1-1[114X).[133X
  
  [8X Casey Donoven and Rhiannon Dougall [108X
        [33X[0;6Yfor  their  contribution  to  the  development  of  the algorithms for
        maximal   subsemigroups   and   smaller   degree  partial  permutation
        representations.[133X
  
  [8X James East [108X
        [33X[0;6Ywho  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.[133X
  
  [8X Zak Mesyan [108X
        [33X[0;6Ywho  contributed to the code for graph inverse semigroups; see Section
        [14X7.10[114X.[133X
  
  [8X Yann Péresse and Yanhui Wang [108X
        [33X[0;6Ywho contributed to the attribute [2XMunnSemigroup[102X ([14X7.2-1[114X).[133X
  
  [8X Jhevon Smith and Ben Steinberg [108X
        [33X[0;6Ywho    contributed   the   function   [2XCharacterTableOfInverseSemigroup[102X
        ([14X11.15-10[114X).[133X
  
  [33X[0;0YWe  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.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (Semigroups)[101X
  
  1 [33X[0;0YThe [5XSemigroups[105X package[133X
    1.1 [33X[0;0YIntroduction[133X
    1.2 [33X[0;0YOverview[133X
  2 [33X[0;0YInstalling [5XSemigroups[105X[133X
    2.1 [33X[0;0YFor those in a hurry[133X
    2.2 [33X[0;0YCompiling the kernel module[133X
    2.3 [33X[0;0YRebuilding the documentation[133X
    2.4 [33X[0;0YTesting your installation[133X
      2.4-1 SemigroupsTestInstall
      2.4-2 SemigroupsTestStandard
      2.4-3 SemigroupsTestExtreme
      2.4-4 SemigroupsTestAll
    2.5 [33X[0;0YMore information during a computation[133X
      2.5-1 InfoSemigroups
  3 [33X[0;0YBipartitions and blocks[133X
    3.1 [33X[0;0YThe family and categories of bipartitions[133X
      3.1-1 IsBipartition
      3.1-2 IsBipartitionCollection
    3.2 [33X[0;0YCreating bipartitions[133X
      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 [33X[0;0YChanging the representation of a bipartition[133X
      3.3-1 AsBipartition
      3.3-2 AsBlockBijection
      3.3-3 AsTransformation
      3.3-4 AsPartialPerm
      3.3-5 AsPermutation
    3.4 [33X[0;0YOperators for bipartitions[133X
      3.4-1 PartialPermLeqBipartition
      3.4-2 NaturalLeqPartialPermBipartition
      3.4-3 NaturalLeqBlockBijection
      3.4-4 PermLeftQuoBipartition
    3.5 [33X[0;0YAttributes for bipartitons[133X
      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 [33X[0;0YCreating blocks and their attributes[133X
      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 [33X[0;0YActions on blocks[133X
      3.7-1 OnRightBlocks
      3.7-2 OnLeftBlocks
    3.8 [33X[0;0YSemigroups of bipartitions[133X
      3.8-1 IsBipartitionSemigroup
      3.8-2 IsBlockBijectionSemigroup
      3.8-3 IsPartialPermBipartitionSemigroup
      3.8-4 IsPermBipartitionGroup
      3.8-5 DegreeOfBipartitionSemigroup
  4 [33X[0;0YPartitioned binary relations (PBRs)[133X
    4.1 [33X[0;0YThe family and categories of PBRs[133X
      4.1-1 IsPBR
      4.1-2 IsPBRCollection
    4.2 [33X[0;0YCreating PBRs[133X
      4.2-1 PBR
      4.2-2 RandomPBR
      4.2-3 EmptyPBR
      4.2-4 IdentityPBR
      4.2-5 UniversalPBR
    4.3 [33X[0;0YChanging the representation of a PBR[133X
      4.3-1 AsPBR
      4.3-2 AsTransformation
      4.3-3 AsPartialPerm
      4.3-4 AsPermutation
    4.4 [33X[0;0YOperators for PBRs[133X
    4.5 [33X[0;0YAttributes for PBRs[133X
      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 [33X[0;0YSemigroups of PBRs[133X
      4.6-1 IsPBRSemigroup
      4.6-2 DegreeOfPBRSemigroup
  5 [33X[0;0YMatrices over semirings[133X
    5.1 [33X[0;0YCreating matrices over semirings[133X
      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 [33X[0;0YMatrix filters[133X
      5.1-9 [33X[0;0YMatrix collection filters[133X
      5.1-10 AsList
      5.1-11 ThresholdTropicalMatrix
      5.1-12 ThresholdNTPMatrix
    5.2 [33X[0;0YOperators for matrices over semirings[133X
    5.3 [33X[0;0YBoolean matrices[133X
      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 [33X[0;0YMatrices over finite fields[133X
      5.4-1 RowSpaceBasis
      5.4-2 RightInverse
    5.5 [33X[0;0YMatrices over the integers[133X
      5.5-1 InverseOp
      5.5-2 IsTorsion
      5.5-3 Order
    5.6 [33X[0;0YMax-plus and min-plus matrices[133X
      5.6-1 InverseOp
      5.6-2 RadialEigenvector
      5.6-3 SpectralRadius
      5.6-4 UnweightedPrecedenceDigraph
    5.7 [33X[0;0YMatrix semigroups[133X
      5.7-1 [33X[0;0YMatrix semigroup filters[133X
      5.7-2 [33X[0;0YMatrix monoid filters[133X
      5.7-3 IsFinite
      5.7-4 IsTorsion
      5.7-5 NormalizeSemigroup
  6 [33X[0;0YSemigroups and monoids defined by generating sets[133X
    6.1 [33X[0;0YUnderlying algorithms[133X
      6.1-1 [33X[0;0YActing semigroups[133X
      6.1-2 IsActingSemigroup
      6.1-3 [33X[0;0YThe Froidure-Pin Algorithm[133X
      6.1-4 CanUseFroidurePin
    6.2 [33X[0;0YSemigroups represented by generators[133X
      6.2-1 InverseMonoidByGenerators
    6.3 [33X[0;0YOptions when creating semigroups[133X
      6.3-1 SEMIGROUPS.DefaultOptionsRec
    6.4 [33X[0;0YSubsemigroups and supersemigroups[133X
      6.4-1 ClosureSemigroup
      6.4-2 SubsemigroupByProperty
      6.4-3 InverseSubsemigroupByProperty
    6.5 [33X[0;0YChanging the representation of a semigroup[133X
      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
      6.5-10 EmbeddingFpMonoid
    6.6 [33X[0;0YRandom semigroups[133X
      6.6-1 RandomSemigroup
  7 [33X[0;0YStandard examples[133X
    7.1 [33X[0;0YTransformation semigroups[133X
      7.1-1 CatalanMonoid
      7.1-2 EndomorphismsPartition
      7.1-3 PartialTransformationMonoid
      7.1-4 SingularTransformationSemigroup
      7.1-5 [33X[0;0YSemigroups of order-preserving transformations[133X
      7.1-6 EndomorphismMonoid
    7.2 [33X[0;0YSemigroups of partial permutations[133X
      7.2-1 MunnSemigroup
      7.2-2 RookMonoid
      7.2-3 [33X[0;0YInverse monoids of order-preserving partial permutations[133X
    7.3 [33X[0;0YSemigroups of bipartitions[133X
      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 [33X[0;0YStandard PBR semigroups[133X
      7.4-1 FullPBRMonoid
    7.5 [33X[0;0YSemigroups of matrices over a finite field[133X
      7.5-1 FullMatrixMonoid
      7.5-2 SpecialLinearMonoid
      7.5-3 IsFullMatrixMonoid
    7.6 [33X[0;0YSemigroups of boolean matrices[133X
      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 [33X[0;0YSemigroups of matrices over a semiring[133X
      7.7-1 FullTropicalMaxPlusMonoid
      7.7-2 FullTropicalMinPlusMonoid
    7.8 [33X[0;0YExamples in various representations[133X
      7.8-1 TrivialSemigroup
      7.8-2 MonogenicSemigroup
      7.8-3 RectangularBand
      7.8-4 FreeSemilattice
      7.8-5 ZeroSemigroup
      7.8-6 LeftZeroSemigroup
      7.8-7 BrandtSemigroup
    7.9 [33X[0;0YFree bands[133X
      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 [33X[0;0YOperators[133X
    7.10 [33X[0;0YGraph inverse semigroups[133X
      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 [33X[0;0YFree inverse semigroups[133X
      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 [33X[0;0YDisplaying free inverse semigroup elements[133X
      7.11-9 [33X[0;0YOperators for free inverse semigroup elements[133X
  8 [33X[0;0YStandard constructions[133X
    8.1 [33X[0;0YProducts of semigroups[133X
      8.1-1 DirectProduct
      8.1-2 WreathProduct
    8.2 [33X[0;0YDual semigroups[133X
      8.2-1 DualSemigroup
      8.2-2 IsDualSemigroupRep
      8.2-3 IsDualSemigroupElement
      8.2-4 AntiIsomorphismDualSemigroup
    8.3 [33X[0;0YStrong semilattices of semigroups[133X
      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 [33X[0;0YMcAlister triple semigroups[133X
      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 [33X[0;0YIdeals[133X
    9.1 [33X[0;0YCreating ideals[133X
      9.1-1 SemigroupIdeal
      9.1-2 Ideals
    9.2 [33X[0;0YAttributes of ideals[133X
      9.2-1 GeneratorsOfSemigroupIdeal
      9.2-2 MinimalIdealGeneratingSet
      9.2-3 SupersemigroupOfIdeal
  10 [33X[0;0YGreen's relations[133X
    10.1 [33X[0;0YCreating Green's classes and representatives[133X
      10.1-1 [33X[0;0YXClassOfYClass[133X
      10.1-2 [33X[0;0YGreensXClassOfElement[133X
      10.1-3 [33X[0;0YGreensXClassOfElementNC[133X
      10.1-4 [33X[0;0YGreensXClasses[133X
      10.1-5 [33X[0;0YXClassReps[133X
      10.1-6 MinimalDClass
      10.1-7 [33X[0;0YMaximalXClasses[133X
      10.1-8 NrRegularDClasses
      10.1-9 [33X[0;0YNrXClasses[133X
      10.1-10 [33X[0;0YPartialOrderOfXClasses[133X
      10.1-11 LengthOfLongestDClassChain
      10.1-12 IsGreensDGreaterThanFunc
    10.2 [33X[0;0YIterators and enumerators of classes and representatives[133X
      10.2-1 [33X[0;0YIteratorOfXClassReps[133X
      10.2-2 [33X[0;0YIteratorOfXClasses[133X
    10.3 [33X[0;0YProperties of Green's classes[133X
      10.3-1 [33X[0;0YLess than for Green's classes[133X
      10.3-2 IsRegularGreensClass
      10.3-3 IsGreensClassNC
    10.4 [33X[0;0YAttributes of Green's classes[133X
      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
    10.5 [33X[0;0YOperations for Green's relations and classes[133X
      10.5-1 LeftGreensMultiplier
  11 [33X[0;0YAttributes and operations for semigroups[133X
    11.1 [33X[0;0YAccessing the elements of a semigroup[133X
      11.1-1 AsListCanonical
      11.1-2 PositionCanonical
      11.1-3 Enumerate
      11.1-4 IsEnumerated
    11.2 [33X[0;0YCayley graphs[133X
      11.2-1 RightCayleyDigraph
    11.3 [33X[0;0YRandom elements of a semigroup[133X
      11.3-1 Random
    11.4 [33X[0;0YProperties of elements in a semigroup[133X
      11.4-1 IndexPeriodOfSemigroupElement
      11.4-2 SmallestIdempotentPower
    11.5 [33X[0;0YOperations for elements in a semigroup[133X
      11.5-1 OneInverseOfSemigroupElement
    11.6 [33X[0;0YExpressing semigroup elements as words in generators[133X
      11.6-1 EvaluateWord
      11.6-2 Factorization
      11.6-3 MinimalFactorization
      11.6-4 NonTrivialFactorization
    11.7 [33X[0;0YGenerating sets[133X
      11.7-1 Generators
      11.7-2 SmallGeneratingSet
      11.7-3 IrredundantGeneratingSubset
      11.7-4 MinimalSemigroupGeneratingSet
      11.7-5 GeneratorsSmallest
      11.7-6 IndecomposableElements
    11.8 [33X[0;0YMinimal ideals and multiplicative zeros[133X
      11.8-1 MinimalIdeal
      11.8-2 RepresentativeOfMinimalIdeal
      11.8-3 MultiplicativeZero
      11.8-4 UnderlyingSemigroupOfSemigroupWithAdjoinedZero
    11.9 [33X[0;0YGroup of units and identity elements[133X
      11.9-1 GroupOfUnits
    11.10 [33X[0;0YIdempotents[133X
      11.10-1 Idempotents
      11.10-2 NrIdempotents
      11.10-3 IdempotentGeneratedSubsemigroup
    11.11 [33X[0;0YMaximal subsemigroups[133X
      11.11-1 MaximalSubsemigroups
      11.11-2 NrMaximalSubsemigroups
      11.11-3 IsMaximalSubsemigroup
    11.12 [33X[0;0YAttributes of transformations and transformation semigroups[133X
      11.12-1 ComponentRepsOfTransformationSemigroup
      11.12-2 ComponentsOfTransformationSemigroup
      11.12-3 CyclesOfTransformationSemigroup
      11.12-4 DigraphOfAction
      11.12-5 DigraphOfActionOnPoints
      11.12-6 FixedPointsOfTransformationSemigroup
      11.12-7 IsTransitive
      11.12-8 SmallestElementSemigroup
      11.12-9 CanonicalTransformation
      11.12-10 IsConnectedTransformationSemigroup
    11.13 [33X[0;0YAttributes of partial perm semigroups[133X
      11.13-1 ComponentRepsOfPartialPermSemigroup
      11.13-2 ComponentsOfPartialPermSemigroup
      11.13-3 CyclesOfPartialPerm
      11.13-4 CyclesOfPartialPermSemigroup
    11.14 [33X[0;0YAttributes of Rees (0-)matrix semigroups[133X
      11.14-1 RZMSDigraph
      11.14-2 RZMSConnectedComponents
    11.15 [33X[0;0YAttributes of inverse semigroups[133X
      11.15-1 NaturalLeqInverseSemigroup
      11.15-2 JoinIrreducibleDClasses
      11.15-3 MajorantClosure
      11.15-4 Minorants
      11.15-5 PrimitiveIdempotents
      11.15-6 RightCosetsOfInverseSemigroup
      11.15-7 SameMinorantsSubgroup
      11.15-8 SmallerDegreePartialPermRepresentation
      11.15-9 VagnerPrestonRepresentation
      11.15-10 CharacterTableOfInverseSemigroup
      11.15-11 EUnitaryInverseCover
    11.16 [33X[0;0YNambooripad partial order[133X
      11.16-1 NambooripadLeqRegularSemigroup
      11.16-2 NambooripadPartialOrder
  12 [33X[0;0YProperties of semigroups[133X
    12.1 [33X[0;0YArbitrary semigroups[133X
      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 [33X[0;0YIsIdempotentGenerated[133X
      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 [33X[0;0YIsXTrivial[133X
      12.1-20 IsSemigroupWithAdjoinedZero
      12.1-21 IsSemilattice
      12.1-22 [33X[0;0YIsSimpleSemigroup[133X
      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 [33X[0;0YInverse semigroups[133X
      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 [33X[0;0YCongruences[133X
    13.1 [33X[0;0YSemigroup congruence objects[133X
      13.1-1 IsSemigroupCongruence
      13.1-2 IsLeftSemigroupCongruence
      13.1-3 IsRightSemigroupCongruence
    13.2 [33X[0;0YCreating congruences[133X
      13.2-1 SemigroupCongruence
      13.2-2 LeftSemigroupCongruence
      13.2-3 RightSemigroupCongruence
    13.3 [33X[0;0YCongruence classes[133X
      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 [33X[0;0YFinding the congruences of a semigroup[133X
      13.4-1 CongruencesOfSemigroup
      13.4-2 MinimalCongruencesOfSemigroup
      13.4-3 PrincipalCongruencesOfSemigroup
      13.4-4 IsCongruencePoset
      13.4-5 LatticeOfCongruences
      13.4-6 CayleyDigraphOfCongruences
      13.4-7 PosetOfPrincipalCongruences
      13.4-8 CongruencesOfPoset
      13.4-9 UnderlyingSemigroupOfCongruencePoset
      13.4-10 PosetOfCongruences
      13.4-11 JoinSemilatticeOfCongruences
      13.4-12 GeneratingCongruencesOfJoinSemilattice
      13.4-13 MinimalCongruences
      13.4-14 NumberOfRightCongruences
      13.4-15 IteratorOfRightCongruences
    13.5 [33X[0;0YComparing congruences[133X
      13.5-1 IsSubrelation
      13.5-2 IsSuperrelation
      13.5-3 MeetSemigroupCongruences
      13.5-4 JoinSemigroupCongruences
    13.6 [33X[0;0YCongruences on Rees matrix semigroups[133X
      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 [33X[0;0YCongruences on inverse semigroups[133X
      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 [33X[0;0YCongruences on graph inverse semigroups[133X
      13.8-1 IsCongruenceByWangPair
      13.8-2 CongruenceByWangPair
      13.8-3 AsCongruenceByWangPair
      13.8-4 GeneratingCongruencesOfLattice
    13.9 [33X[0;0YRees congruences[133X
      13.9-1 SemigroupIdealOfReesCongruence
      13.9-2 IsReesCongruenceClass
    13.10 [33X[0;0YUniversal and trivial congruences[133X
      13.10-1 IsUniversalSemigroupCongruence
      13.10-2 IsUniversalSemigroupCongruenceClass
      13.10-3 UniversalSemigroupCongruence
      13.10-4 TrivialCongruence
  14 [33X[0;0YSemigroup homomorphisms[133X
    14.1 [33X[0;0YHomomorphisms of arbitrary semigroups[133X
      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 [33X[0;0YIsomorphisms of arbitrary semigroups[133X
      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 [33X[0;0YIsomorphisms of Rees (0-)matrix semigroups[133X
      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 [33X[0;0YOperators for isomorphisms of Rees (0-)matrix semigroups[133X
  15 [33X[0;0YFinitely presented semigroups and Tietze transformations[133X
    15.1 [33X[0;0YChanging representation for words and strings[133X
      15.1-1 WordToString
      15.1-2 RandomWord
      15.1-3 StandardiseWord
      15.1-4 StringToWord
    15.2 [33X[0;0YHelper functions[133X
      15.2-1 ParseRelations
      15.2-2 ElementOfFpSemigroup
      15.2-3 ElementOfFpMonoid
      15.2-4 FreeMonoidAndAssignGeneratorVars
      15.2-5 IsSubsemigroupOfFpMonoid
    15.3 [33X[0;0YCreating Tietze transformation objects[133X
      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 [33X[0;0YPrinting Tietze transformation objects[133X
      15.4-1 StzPrintRelations
      15.4-2 StzPrintRelation
      15.4-3 StzPrintGenerators
      15.4-4 StzPrintPresentation
    15.5 [33X[0;0YChanging Tietze transformation objects[133X
      15.5-1 StzAddRelation
      15.5-2 StzRemoveRelation
      15.5-3 StzAddGenerator
      15.5-4 StzRemoveGenerator
      15.5-5 StzSubstituteRelation
    15.6 [33X[0;0YConverting a Tietze transformation object into a fp semigroup[133X
      15.6-1 TietzeForwardMap
      15.6-2 TietzeBackwardMap
      15.6-3 StzIsomorphism
    15.7 [33X[0;0YAutomatically simplifying a Tietze transformation object[133X
      15.7-1 StzSimplifyOnce
      15.7-2 StzSimplifyPresentation
    15.8 [33X[0;0YAutomatically simplifying an fp semigroup[133X
      15.8-1 SimplifyFpSemigroup
      15.8-2 SimplifiedFpSemigroup
      15.8-3 UnreducedFpSemigroup
      15.8-4 FpTietzeIsomorphism
  16 [33X[0;0YVisualising semigroups and elements[133X
    16.1 [33X[0;0Y[10Xdot[110X pictures[133X
      16.1-1 DotString
      16.1-2 DotString
      16.1-3 DotSemilatticeOfIdempotents
      16.1-4 DotLeftCayleyDigraph
    16.2 [33X[0;0Y[10Xtex[110X output[133X
      16.2-1 TexString
    16.3 [33X[0;0Y[10Xtikz[110X pictures[133X
      16.3-1 TikzString
      16.3-2 TikzLeftCayleyDigraph
  17 [33X[0;0YIO[133X
    17.1 [33X[0;0YReading and writing elements to a file[133X
      17.1-1 ReadGenerators
      17.1-2 WriteGenerators
      17.1-3 IteratorFromGeneratorsFile
    17.2 [33X[0;0YReading and writing multiplication tables to a file[133X
      17.2-1 ReadMultiplicationTable
      17.2-2 WriteMultiplicationTable
      17.2-3 IteratorFromMultiplicationTableFile
  18 [33X[0;0YTranslations[133X
    18.1 [33X[0;0YMethods for translations[133X
      18.1-1 [33X[0;0YIsXTranslation[133X
      18.1-2 IsBitranslation
      18.1-3 [33X[0;0YIsXTranslationCollection[133X
      18.1-4 [33X[0;0YXPartOfBitranslation[133X
      18.1-5 [33X[0;0YXTranslation[133X
      18.1-6 Bitranslation
      18.1-7 [33X[0;0YUnderlyingSemigroup[133X
      18.1-8 [33X[0;0YXTranslationsSemigroupOfFamily[133X
      18.1-9 [33X[0;0YTypeXTranslationSemigroupElements[133X
      18.1-10 [33X[0;0YXTranslations[133X
      18.1-11 TranslationalHull
      18.1-12 [33X[0;0YNrXTranslations[133X
      18.1-13 [33X[0;0YInnerXTranslations[133X
      18.1-14 InnerTranslationalHull
      18.1-15 UnderlyingRepresentatives
      18.1-16 ImageSetOfTranslation
  
  
  [32X
