  
  
                                   [1X[5XSemigroups[105X[101X
  
  
                                 Version 3.4.0
  
  
                                 J. D. Mitchell
  
                                 Stuart Burrell
  
                                 Manuel Delgado
  
                                   James East
  
                                Attila Egri-Nagy
  
                                  Luke Elliott
  
                                  Nicholas Ham
  
                                    Max Horn
  
                                Chris Jefferson
  
                                 Dima Pasechnik
  
                                 Julius Jonušas
  
                                Markus Pfeiffer
  
                                 Chris Russell
  
                                 Ben Steinberg
  
                                   Finn Smith
  
                                  Jhevon Smith
  
                                 Michael Torpey
  
                                  Murray Whyte
  
                                 Wilf A. Wilson
  
  
  
  J. D. Mitchell
      Email:    [7Xmailto:jdm3@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://jdbm.me[107X
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  [5XSemigroups[105X  package is a [5XGAP[105X package containing methods for semigroups,
  monoids,  and  inverse  semigroups. There are particularly efficient methods
  for   semigroups   or   ideals   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.[133X
  
  [33X[0;0Y[5XSemigroups[105X  contains efficient methods 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  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 variety of further properties.[133X
  
  [33X[0;0YThere are methods for finding presentations for a semigroup, the congruences
  of  a  semigroup,  the normalizer of a semigroup in a permutation group, 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© 2011-20 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
  ([7Xhttp://www.fsf.org/licenses/gpl.html[107X)  as  published  by  the Free Software
  Foundation;  either  version 3 of the License, or (at your option) any later
  version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YI  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.[133X
  
  [33X[0;0YStuart  Burrell contributed methods for checking finiteness of semigroups of
  matrices of the max-plus and min-plus semirings.[133X
  
  [33X[0;0YManuel  Delgado  and  Attila Egri-Nagy contributed to the function [2XDotString[102X
  ([14X19.1-1[114X).[133X
  
  [33X[0;0YJames 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.[133X
  
  [33X[0;0YNick   Ham   contributed  many  of  the  standard  examples  of  bipartition
  semigroups.[133X
  
  [33X[0;0YMax  Horn  contributed  many patches and fixes, in particular, to the kernel
  module.[133X
  
  [33X[0;0YChris Jefferson contributed several patches and fixes to the build system.[133X
  
  [33X[0;0YJulius  Jonušas contributed the part of the package relating to free inverse
  semigroups, and contributed to the code for ideals.[133X
  
  [33X[0;0YZak Mesyan contributed to the code for graph inverse semigroups; see Chapter
  [14X11[114X.[133X
  
  [33X[0;0YDima Pasechnik contributed to the build system of the kernel module.[133X
  
  [33X[0;0YMarkus  Pfeiffer contributed the majority of the code relating to semigroups
  of matrices over finite fields.[133X
  
  [33X[0;0YYann  Péresse  and  Yanhui  Wang  contributed to the attribute [2XMunnSemigroup[102X
  ([14X8.2-1[114X).[133X
  
  [33X[0;0YJhevon    Smith    and    Ben    Steinberg    contributed    the    function
  [2XCharacterTableOfInverseSemigroup[102X ([14X16.1-10[114X).[133X
  
  [33X[0;0YMichael Torpey contributed the part of the package relating to congruences.[133X
  
  [33X[0;0YMurray Whyte was kind enough to update the bibliography in 2019.[133X
  
  [33X[0;0YWilf  A.  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.[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 scholarship of M. Torpey when he worked
  on this project (EP/M506631/1).[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;0YPackage dependencies[133X
    2.3 [33X[0;0YCompiling the kernel module[133X
    2.4 [33X[0;0YRebuilding the documentation[133X
      2.4-1 SemigroupsMakeDoc
    2.5 [33X[0;0YTesting your installation[133X
      2.5-1 SemigroupsTestInstall
      2.5-2 SemigroupsTestStandard
      2.5-3 SemigroupsTestExtreme
      2.5-4 SemigroupsTestAll
    2.6 [33X[0;0YMore information during a computation[133X
      2.6-1 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 BlocksNC
      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 NewMatrixOverFiniteField
      5.4-2 IdentityMatrixOverFiniteField
      5.4-3 NewIdentityMatrixOverFiniteField
      5.4-4 RowSpaceBasis
      5.4-5 RowRank
      5.4-6 RightInverse
      5.4-7 BaseDomain
      5.4-8 TransposedMatImmutable
    5.5 [33X[0;0YInteger Matrices[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
      5.7-6 [33X[0;0YMatrix groups[133X
      5.7-7 IsMatrixOverFiniteFieldGroup
      5.7-8 \^
      5.7-9 IsomorphismMatrixGroup
      5.7-10 AsMatrixGroup
  6 [33X[0;0YCreating semigroups and monoids[133X
    6.1 [33X[0;0YUnderlying algorithms and related representations[133X
      6.1-1 [33X[0;0YActing semigroups[133X
      6.1-2 [33X[0;0YEnumerable semigroups[133X
      6.1-3 IsActingSemigroup
      6.1-4 IsEnumerableSemigroupRep
    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;0YNew semigroups from old[133X
      6.4-1 ClosureSemigroup
      6.4-2 SubsemigroupByProperty
      6.4-3 InverseSubsemigroupByProperty
      6.4-4 DirectProduct
      6.4-5 WreathProduct
    6.5 [33X[0;0YDual semigroups[133X
      6.5-1 DualSemigroup
      6.5-2 IsDualSemigroupRep
      6.5-3 IsDualSemigroupElement
      6.5-4 AntiIsomorphismDualSemigroup
    6.6 [33X[0;0YChanging the representation of a semigroup[133X
      6.6-1 IsomorphismSemigroup
      6.6-2 IsomorphismMonoid
      6.6-3 AsSemigroup
      6.6-4 AsMonoid
      6.6-5 IsomorphismPermGroup
      6.6-6 RZMSNormalization
      6.6-7 RMSNormalization
    6.7 [33X[0;0YRandom semigroups[133X
      6.7-1 RandomSemigroup
    6.8 [33X[0;0YEndomorphism monoid of a digraph[133X
      6.8-1 EndomorphismMonoid
  7 [33X[0;0YIdeals[133X
    7.1 [33X[0;0YCreating ideals[133X
      7.1-1 SemigroupIdeal
      7.1-2 Ideals
    7.2 [33X[0;0YAttributes of ideals[133X
      7.2-1 GeneratorsOfSemigroupIdeal
      7.2-2 MinimalIdealGeneratingSet
      7.2-3 SupersemigroupOfIdeal
  8 [33X[0;0YStandard examples[133X
    8.1 [33X[0;0YTransformation semigroups[133X
      8.1-1 CatalanMonoid
      8.1-2 EndomorphismsPartition
      8.1-3 PartialTransformationMonoid
      8.1-4 SingularTransformationSemigroup
      8.1-5 [33X[0;0YSemigroups of order-preserving transformations[133X
    8.2 [33X[0;0YSemigroups of partial permutations[133X
      8.2-1 MunnSemigroup
      8.2-2 RookMonoid
      8.2-3 [33X[0;0YInverse monoids of order-preserving partial permutations[133X
    8.3 [33X[0;0YSemigroups of bipartitions[133X
      8.3-1 PartitionMonoid
      8.3-2 BrauerMonoid
      8.3-3 JonesMonoid
      8.3-4 PartialJonesMonoid
      8.3-5 AnnularJonesMonoid
      8.3-6 MotzkinMonoid
      8.3-7 DualSymmetricInverseSemigroup
      8.3-8 UniformBlockBijectionMonoid
      8.3-9 PlanarPartitionMonoid
      8.3-10 ModularPartitionMonoid
      8.3-11 ApsisMonoid
    8.4 [33X[0;0YStandard PBR semigroups[133X
      8.4-1 FullPBRMonoid
    8.5 [33X[0;0YSemigroups of matrices over a finite field[133X
      8.5-1 FullMatrixMonoid
      8.5-2 SpecialLinearMonoid
      8.5-3 IsFullMatrixMonoid
    8.6 [33X[0;0YSemigroups of boolean matrices[133X
      8.6-1 FullBooleanMatMonoid
      8.6-2 RegularBooleanMatMonoid
      8.6-3 ReflexiveBooleanMatMonoid
      8.6-4 HallMonoid
      8.6-5 GossipMonoid
      8.6-6 TriangularBooleanMatMonoid
    8.7 [33X[0;0YSemigroups of matrices over a semiring[133X
      8.7-1 FullTropicalMaxPlusMonoid
      8.7-2 FullTropicalMinPlusMonoid
  9 [33X[0;0YStandard constructions[133X
    9.1 [33X[0;0YStandard constructions[133X
      9.1-1 TrivialSemigroup
      9.1-2 MonogenicSemigroup
      9.1-3 RectangularBand
      9.1-4 ZeroSemigroup
      9.1-5 LeftZeroSemigroup
      9.1-6 BrandtSemigroup
  10 [33X[0;0YFree objects[133X
    10.1 [33X[0;0YFree inverse semigroups[133X
      10.1-1 FreeInverseSemigroup
      10.1-2 IsFreeInverseSemigroupCategory
      10.1-3 IsFreeInverseSemigroup
      10.1-4 IsFreeInverseSemigroupElement
      10.1-5 IsFreeInverseSemigroupElementCollection
    10.2 [33X[0;0YDisplaying free inverse semigroup elements[133X
    10.3 [33X[0;0YOperators and operations for free inverse semigroup elements[133X
      10.3-1 CanonicalForm
      10.3-2 MinimalWord
    10.4 [33X[0;0YFree bands[133X
      10.4-1 FreeBand
      10.4-2 IsFreeBandCategory
      10.4-3 IsFreeBand
      10.4-4 IsFreeBandElement
      10.4-5 IsFreeBandElementCollection
      10.4-6 IsFreeBandSubsemigroup
      10.4-7 ContentOfFreeBandElement
    10.5 [33X[0;0YOperators and operations for free band elements[133X
      10.5-1 GreensDClassOfElement
  11 [33X[0;0YGraph inverse semigroups[133X
    11.1 [33X[0;0YCreating graph inverse semigroups[133X
      11.1-1 GraphInverseSemigroup
      11.1-2 Range
      11.1-3 IsVertex
      11.1-4 IsGraphInverseSemigroup
      11.1-5 GraphOfGraphInverseSemigroup
      11.1-6 IsGraphInverseSemigroupElementCollection
      11.1-7 IsGraphInverseSubsemigroup
  12 [33X[0;0YMcAlister triple semigroups and E-unitary inverse semigroups[133X
    12.1 [33X[0;0YCreating McAlister triple semigroups[133X
      12.1-1 IsMcAlisterTripleSemigroup
      12.1-2 McAlisterTripleSemigroup
      12.1-3 McAlisterTripleSemigroupGroup
      12.1-4 McAlisterTripleSemigroupPartialOrder
      12.1-5 McAlisterTripleSemigroupSemilattice
      12.1-6 McAlisterTripleSemigroupAction
      12.1-7 IsMcAlisterTripleSemigroupElement
      12.1-8 McAlisterTripleSemigroupElement
  13 [33X[0;0YGreen's relations[133X
    13.1 [33X[0;0YCreating Green's classes and representatives[133X
      13.1-1 [33X[0;0YXClassOfYClass[133X
      13.1-2 [33X[0;0YGreensXClassOfElement[133X
      13.1-3 [33X[0;0YGreensXClassOfElementNC[133X
      13.1-4 [33X[0;0YGreensXClasses[133X
      13.1-5 [33X[0;0YXClassReps[133X
      13.1-6 MinimalDClass
      13.1-7 MaximalDClasses
      13.1-8 NrRegularDClasses
      13.1-9 [33X[0;0YNrXClasses[133X
      13.1-10 PartialOrderOfDClasses
      13.1-11 LengthOfLongestDClassChain
      13.1-12 IsGreensDGreaterThanFunc
    13.2 [33X[0;0YIterators and enumerators of classes and representatives[133X
      13.2-1 [33X[0;0YIteratorOfXClassReps[133X
      13.2-2 [33X[0;0YIteratorOfXClasses[133X
    13.3 [33X[0;0YProperties of Green's classes[133X
      13.3-1 [33X[0;0YLess than for Green's classes[133X
      13.3-2 IsRegularGreensClass
      13.3-3 IsGreensClassNC
    13.4 [33X[0;0YAttributes of Green's classes[133X
      13.4-1 GroupHClass
      13.4-2 SchutzenbergerGroup
      13.4-3 StructureDescriptionSchutzenbergerGroups
      13.4-4 StructureDescriptionMaximalSubgroups
      13.4-5 MultiplicativeNeutralElement
      13.4-6 StructureDescription
      13.4-7 InjectionPrincipalFactor
      13.4-8 PrincipalFactor
  14 [33X[0;0YAttributes and operations for semigroups[133X
    14.1 [33X[0;0YAccessing the elements of a semigroup[133X
      14.1-1 AsListCanonical
      14.1-2 PositionCanonical
      14.1-3 Enumerate
      14.1-4 IsFullyEnumerated
    14.2 [33X[0;0YCayley graphs[133X
      14.2-1 RightCayleyDigraph
    14.3 [33X[0;0YRandom elements of a semigroup[133X
      14.3-1 Random
    14.4 [33X[0;0YProperties of elements in a semigroup[133X
      14.4-1 IndexPeriodOfSemigroupElement
      14.4-2 SmallestIdempotentPower
    14.5 [33X[0;0YExpressing semigroup elements as words in generators[133X
      14.5-1 EvaluateWord
      14.5-2 Factorization
      14.5-3 MinimalFactorization
      14.5-4 NonTrivialFactorization
    14.6 [33X[0;0YGenerating sets[133X
      14.6-1 Generators
      14.6-2 SmallGeneratingSet
      14.6-3 IrredundantGeneratingSubset
      14.6-4 MinimalSemigroupGeneratingSet
      14.6-5 GeneratorsSmallest
      14.6-6 IndecomposableElements
    14.7 [33X[0;0YMinimal ideals and multiplicative zeros[133X
      14.7-1 MinimalIdeal
      14.7-2 RepresentativeOfMinimalIdeal
      14.7-3 MultiplicativeZero
      14.7-4 UnderlyingSemigroupOfSemigroupWithAdjoinedZero
    14.8 [33X[0;0YGroup of units and identity elements[133X
      14.8-1 GroupOfUnits
    14.9 [33X[0;0YIdempotents[133X
      14.9-1 Idempotents
      14.9-2 NrIdempotents
      14.9-3 IdempotentGeneratedSubsemigroup
    14.10 [33X[0;0YMaximal subsemigroups[133X
      14.10-1 MaximalSubsemigroups
      14.10-2 NrMaximalSubsemigroups
      14.10-3 IsMaximalSubsemigroup
    14.11 [33X[0;0YThe normalizer of a semigroup[133X
      14.11-1 Normalizer
    14.12 [33X[0;0YAttributes of transformations and transformation semigroups[133X
      14.12-1 ComponentRepsOfTransformationSemigroup
      14.12-2 ComponentsOfTransformationSemigroup
      14.12-3 CyclesOfTransformationSemigroup
      14.12-4 DigraphOfActionOnPairs
      14.12-5 DigraphOfActionOnPoints
      14.12-6 FixedPointsOfTransformationSemigroup
      14.12-7 IsTransitive
      14.12-8 SmallestElementSemigroup
      14.12-9 CanonicalTransformation
      14.12-10 IsConnectedTransformationSemigroup
    14.13 [33X[0;0YAttributes of partial perm semigroups[133X
      14.13-1 ComponentRepsOfPartialPermSemigroup
      14.13-2 ComponentsOfPartialPermSemigroup
      14.13-3 CyclesOfPartialPerm
      14.13-4 CyclesOfPartialPermSemigroup
    14.14 [33X[0;0YAttributes of Rees (0-)matrix semigroups[133X
      14.14-1 RZMSDigraph
      14.14-2 RZMSConnectedComponents
    14.15 [33X[0;0YChanging the representation of a semigroup[133X
      14.15-1 IsomorphismReesMatrixSemigroup
    14.16 [33X[0;0YThe Nambooripad partial order of a regular semigroup[133X
      14.16-1 NambooripadLeqRegularSemigroup
      14.16-2 NambooripadPartialOrder
  15 [33X[0;0YProperties of semigroups[133X
    15.1 [33X[0;0YProperties of semigroups[133X
      15.1-1 IsBand
      15.1-2 IsBlockGroup
      15.1-3 IsCommutativeSemigroup
      15.1-4 IsCompletelyRegularSemigroup
      15.1-5 IsCongruenceFreeSemigroup
      15.1-6 IsSurjectiveSemigroup
      15.1-7 IsGroupAsSemigroup
      15.1-8 [33X[0;0YIsIdempotentGenerated[133X
      15.1-9 IsLeftSimple
      15.1-10 IsLeftZeroSemigroup
      15.1-11 IsMonogenicSemigroup
      15.1-12 IsMonogenicMonoid
      15.1-13 IsMonoidAsSemigroup
      15.1-14 IsOrthodoxSemigroup
      15.1-15 IsRectangularBand
      15.1-16 IsRectangularGroup
      15.1-17 IsRegularSemigroup
      15.1-18 IsRightZeroSemigroup
      15.1-19 [33X[0;0YIsXTrivial[133X
      15.1-20 IsSemigroupWithAdjoinedZero
      15.1-21 IsSemilattice
      15.1-22 [33X[0;0YIsSimpleSemigroup[133X
      15.1-23 IsSynchronizingSemigroup
      15.1-24 IsUnitRegularMonoid
      15.1-25 IsZeroGroup
      15.1-26 IsZeroRectangularBand
      15.1-27 IsZeroSemigroup
      15.1-28 IsZeroSimpleSemigroup
  16 [33X[0;0YProperties and attributes specific to inverse semigroups[133X
    16.1 [33X[0;0YAttributes specific to inverse semigroups[133X
      16.1-1 NaturalLeqInverseSemigroup
      16.1-2 JoinIrreducibleDClasses
      16.1-3 MajorantClosure
      16.1-4 Minorants
      16.1-5 PrimitiveIdempotents
      16.1-6 RightCosetsOfInverseSemigroup
      16.1-7 SameMinorantsSubgroup
      16.1-8 SmallerDegreePartialPermRepresentation
      16.1-9 VagnerPrestonRepresentation
      16.1-10 CharacterTableOfInverseSemigroup
      16.1-11 EUnitaryInverseCover
    16.2 [33X[0;0YProperties of inverse semigroups[133X
      16.2-1 IsCliffordSemigroup
      16.2-2 IsBrandtSemigroup
      16.2-3 IsEUnitaryInverseSemigroup
      16.2-4 IsFInverseSemigroup
      16.2-5 IsFInverseMonoid
      16.2-6 IsFactorisableInverseMonoid
      16.2-7 IsJoinIrreducible
      16.2-8 IsMajorantlyClosed
      16.2-9 IsMonogenicInverseSemigroup
      16.2-10 IsMonogenicInverseMonoid
  17 [33X[0;0YCongruences[133X
    17.1 [33X[0;0YSemigroup congruence objects[133X
      17.1-1 IsSemigroupCongruence
      17.1-2 IsLeftSemigroupCongruence
      17.1-3 IsRightSemigroupCongruence
    17.2 [33X[0;0YCreating congruences[133X
      17.2-1 SemigroupCongruence
      17.2-2 LeftSemigroupCongruence
      17.2-3 RightSemigroupCongruence
      17.2-4 GeneratingPairsOfSemigroupCongruence
    17.3 [33X[0;0YCongruence classes[133X
      17.3-1 IsCongruenceClass
      17.3-2 IsLeftCongruenceClass
      17.3-3 IsRightCongruenceClass
      17.3-4 CongruenceClassOfElement
      17.3-5 CongruenceClasses
      17.3-6 NonTrivialEquivalenceClasses
      17.3-7 NonTrivialCongruenceClasses
      17.3-8 NrEquivalenceClasses
      17.3-9 NrCongruenceClasses
      17.3-10 EquivalenceRelationLookup
      17.3-11 EquivalenceRelationCanonicalLookup
      17.3-12 EquivalenceRelationCanonicalPartition
      17.3-13 OnLeftCongruenceClasses
      17.3-14 OnRightCongruenceClasses
    17.4 [33X[0;0YFinding the congruences of a semigroup[133X
      17.4-1 CongruencesOfSemigroup
      17.4-2 MinimalCongruencesOfSemigroup
      17.4-3 PrincipalCongruencesOfSemigroup
      17.4-4 IsCongruencePoset
      17.4-5 LatticeOfCongruences
      17.4-6 PosetOfPrincipalCongruences
      17.4-7 CongruencesOfPoset
      17.4-8 UnderlyingSemigroupOfCongruencePoset
      17.4-9 PosetOfCongruences
      17.4-10 JoinSemilatticeOfCongruences
      17.4-11 MinimalCongruences
    17.5 [33X[0;0YComparing congruences[133X
      17.5-1 IsSubrelation
      17.5-2 IsSuperrelation
      17.5-3 MeetSemigroupCongruences
      17.5-4 JoinSemigroupCongruences
    17.6 [33X[0;0YCongruences on Rees matrix semigroups[133X
      17.6-1 IsRMSCongruenceByLinkedTriple
      17.6-2 RMSCongruenceByLinkedTriple
      17.6-3 IsRMSCongruenceClassByLinkedTriple
      17.6-4 RMSCongruenceClassByLinkedTriple
      17.6-5 IsLinkedTriple
      17.6-6 CanonicalRepresentative
      17.6-7 AsSemigroupCongruenceByGeneratingPairs
      17.6-8 AsRMSCongruenceByLinkedTriple
    17.7 [33X[0;0YCongruences on inverse semigroups[133X
      17.7-1 IsInverseSemigroupCongruenceByKernelTrace
      17.7-2 InverseSemigroupCongruenceByKernelTrace
      17.7-3 AsInverseSemigroupCongruenceByKernelTrace
      17.7-4 KernelOfSemigroupCongruence
      17.7-5 TraceOfSemigroupCongruence
      17.7-6 IsInverseSemigroupCongruenceClassByKernelTrace
      17.7-7 MinimumGroupCongruence
    17.8 [33X[0;0YRees congruences[133X
      17.8-1 SemigroupIdealOfReesCongruence
      17.8-2 IsReesCongruenceClass
    17.9 [33X[0;0YUniversal congruences[133X
      17.9-1 IsUniversalSemigroupCongruence
      17.9-2 IsUniversalSemigroupCongruenceClass
      17.9-3 UniversalSemigroupCongruence
  18 [33X[0;0YSemigroup homomorphisms[133X
    18.1 [33X[0;0YIsomorphisms of arbitrary semigroups[133X
      18.1-1 IsIsomorphicSemigroup
      18.1-2 SmallestMultiplicationTable
      18.1-3 CanonicalMultiplicationTable
      18.1-4 CanonicalMultiplicationTablePerm
      18.1-5 OnMultiplicationTable
      18.1-6 IsomorphismSemigroups
      18.1-7 AutomorphismGroup
    18.2 [33X[0;0YIsomorphisms of Rees (0-)matrix semigroups[133X
      18.2-1 IsRMSIsoByTriple
      18.2-2 RMSIsoByTriple
      18.2-3 ELM_LIST
      18.2-4 CompositionMapping2
      18.2-5 ImagesElm
      18.2-6 CanonicalReesZeroMatrixSemigroup
      18.2-7 [33X[0;0YOperators for isomorphisms of Rees (0-)matrix semigroup by
      triples[133X
  19 [33X[0;0YVisualising semigroups and elements[133X
    19.1 [33X[0;0Y[10Xdot[110X pictures[133X
      19.1-1 DotString
      19.1-2 DotString
      19.1-3 DotSemilatticeOfIdempotents
      19.1-4 DotLeftCayleyDigraph
    19.2 [33X[0;0Y[10Xtex[110X output[133X
      19.2-1 TexString
    19.3 [33X[0;0Y[10Xtikz[110X pictures[133X
      19.3-1 TikzString
      19.3-2 TikzLeftCayleyDigraph
  20 [33X[0;0YIO[133X
    20.1 [33X[0;0YReading and writing elements to a file[133X
      20.1-1 ReadGenerators
      20.1-2 WriteGenerators
      20.1-3 IteratorFromGeneratorsFile
    20.2 [33X[0;0YReading and writing multiplication tables to a file[133X
      20.2-1 ReadMultiplicationTable
      20.2-2 WriteMultiplicationTable
      20.2-3 IteratorFromMultiplicationTableFile
  
  
  [32X
