  
  
                                   [1X[5XSemigroups[105X[101X
  
  
                                 Version 3.0.7
  
  
                                 J. D. Mitchell
  
                                 Manuel Delgado
  
                                   James East
  
                                Attila Egri-Nagy
  
                                  Nicholas Ham
  
                                 Julius Jonušas
  
                                Markus Pfeiffer
  
                                 Ben Steinberg
  
                                  Jhevon Smith
  
                                 Michael Torpey
  
                                 Wilf A. Wilson
  
  
  
  J. D. Mitchell
      Email:    [7Xmailto:jdm3@st-and.ac.uk[107X
      Homepage: [7Xhttp://tinyurl.com/jdmitchell[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© 2011-17 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 functions [2XSplash[102X
  ([14X18.1-1[114X) and [2XDotString[102X ([14X18.2-1[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;0YJulius  Jonušas contributed the part of the package relating to free inverse
  semigroups, and contributed to the code for ideals.[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 ([14X15.1-10[114X).[133X
  
  [33X[0;0YMichael Torpey contributed the part of the package relating to congruences.[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 J. Jonušas and W. 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
    2.6 [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
    5.4 [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
    6.5 [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
    6.6 [33X[0;0YRandom semigroups[133X
      6.6-1 RandomSemigroup
    6.7 [33X[0;0YEndomorphism monoid of a digraph[133X
      6.7-1 EndomorphismMonoid
  7 [33X[0;0YIdeals[133X
    7.1 [33X[0;0YCreating ideals[133X
      7.1-1 SemigroupIdeal
    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
  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;0YGreen's relations[133X
    12.1 [33X[0;0YCreating Green's classes and representatives[133X
      12.1-1 [33X[0;0YXClassOfYClass[133X
      12.1-2 [33X[0;0YGreensXClassOfElement[133X
      12.1-3 [33X[0;0YGreensXClassOfElementNC[133X
      12.1-4 [33X[0;0YGreensXClasses[133X
      12.1-5 [33X[0;0YXClassReps[133X
      12.1-6 MinimalDClass
      12.1-7 MaximalDClasses
      12.1-8 NrRegularDClasses
      12.1-9 [33X[0;0YNrXClasses[133X
      12.1-10 PartialOrderOfDClasses
      12.1-11 LengthOfLongestDClassChain
      12.1-12 IsGreensDGreaterThanFunc
    12.2 [33X[0;0YIterators and enumerators of classes and representatives[133X
      12.2-1 [33X[0;0YIteratorOfXClassReps[133X
      12.2-2 [33X[0;0YIteratorOfXClasses[133X
    12.3 [33X[0;0YProperties of Green's classes[133X
      12.3-1 [33X[0;0YLess than for Green's classes[133X
      12.3-2 IsRegularGreensClass
      12.3-3 IsGreensClassNC
    12.4 [33X[0;0YAttributes of Green's classes[133X
      12.4-1 GroupHClass
      12.4-2 SchutzenbergerGroup
      12.4-3 StructureDescriptionSchutzenbergerGroups
      12.4-4 StructureDescriptionMaximalSubgroups
      12.4-5 MultiplicativeNeutralElement
      12.4-6 StructureDescription
      12.4-7 InjectionPrincipalFactor
      12.4-8 PrincipalFactor
  13 [33X[0;0YAttributes and operations for semigroups[133X
    13.1 [33X[0;0YAccessing the elements of a semigroup[133X
      13.1-1 AsListCanonical
      13.1-2 PositionCanonical
      13.1-3 Enumerate
      13.1-4 IsFullyEnumerated
    13.2 [33X[0;0YCayley graphs[133X
      13.2-1 RightCayleyGraphSemigroup
    13.3 [33X[0;0YRandom elements of a semigroup[133X
      13.3-1 Random
    13.4 [33X[0;0YProperties of elements in a semigroup[133X
      13.4-1 IndexPeriodOfSemigroupElement
      13.4-2 SmallestIdempotentPower
    13.5 [33X[0;0YExpressing semigroup elements as words in generators[133X
      13.5-1 EvaluateWord
      13.5-2 Factorization
      13.5-3 MinimalFactorization
    13.6 [33X[0;0YGenerating sets[133X
      13.6-1 Generators
      13.6-2 SmallGeneratingSet
      13.6-3 IrredundantGeneratingSubset
      13.6-4 MinimalSemigroupGeneratingSet
      13.6-5 GeneratorsSmallest
    13.7 [33X[0;0YMinimal ideals and multiplicative zeros[133X
      13.7-1 MinimalIdeal
      13.7-2 RepresentativeOfMinimalIdeal
      13.7-3 MultiplicativeZero
      13.7-4 UnderlyingSemigroupOfSemigroupWithAdjoinedZero
    13.8 [33X[0;0YGroup of units and identity elements[133X
      13.8-1 GroupOfUnits
    13.9 [33X[0;0YIdempotents[133X
      13.9-1 Idempotents
      13.9-2 NrIdempotents
      13.9-3 IdempotentGeneratedSubsemigroup
    13.10 [33X[0;0YMaximal subsemigroups[133X
      13.10-1 MaximalSubsemigroups
      13.10-2 NrMaximalSubsemigroups
      13.10-3 IsMaximalSubsemigroup
    13.11 [33X[0;0YThe normalizer of a semigroup[133X
      13.11-1 Normalizer
    13.12 [33X[0;0YAttributes of transformations and transformation semigroups[133X
      13.12-1 ComponentRepsOfTransformationSemigroup
      13.12-2 ComponentsOfTransformationSemigroup
      13.12-3 CyclesOfTransformationSemigroup
      13.12-4 DigraphOfActionOnPairs
      13.12-5 DigraphOfActionOnPoints
      13.12-6 FixedPointsOfTransformationSemigroup
      13.12-7 IsTransitive
      13.12-8 SmallestElementSemigroup
      13.12-9 CanonicalTransformation
      13.12-10 IsConnectedTransformationSemigroup
    13.13 [33X[0;0YAttributes of partial perm semigroups[133X
      13.13-1 ComponentRepsOfPartialPermSemigroup
      13.13-2 ComponentsOfPartialPermSemigroup
      13.13-3 CyclesOfPartialPerm
      13.13-4 CyclesOfPartialPermSemigroup
    13.14 [33X[0;0YAttributes of Rees (0-)matrix semigroups[133X
      13.14-1 RZMSDigraph
      13.14-2 RZMSConnectedComponents
    13.15 [33X[0;0YChanging the representation of a semigroup[133X
      13.15-1 IsomorphismReesMatrixSemigroup
  14 [33X[0;0YProperties of semigroups[133X
    14.1 [33X[0;0YProperties of semigroups[133X
      14.1-1 IsBand
      14.1-2 IsBlockGroup
      14.1-3 IsCommutativeSemigroup
      14.1-4 IsCompletelyRegularSemigroup
      14.1-5 IsCongruenceFreeSemigroup
      14.1-6 IsGroupAsSemigroup
      14.1-7 [33X[0;0YIsIdempotentGenerated[133X
      14.1-8 IsLeftSimple
      14.1-9 IsLeftZeroSemigroup
      14.1-10 IsMonogenicSemigroup
      14.1-11 IsMonogenicMonoid
      14.1-12 IsMonoidAsSemigroup
      14.1-13 IsOrthodoxSemigroup
      14.1-14 IsRectangularBand
      14.1-15 IsRectangularGroup
      14.1-16 IsRegularSemigroup
      14.1-17 IsRightZeroSemigroup
      14.1-18 [33X[0;0YIsXTrivial[133X
      14.1-19 IsSemigroupWithAdjoinedZero
      14.1-20 IsSemilattice
      14.1-21 [33X[0;0YIsSimpleSemigroup[133X
      14.1-22 IsSynchronizingSemigroup
      14.1-23 IsUnitRegularMonoid
      14.1-24 IsZeroGroup
      14.1-25 IsZeroRectangularBand
      14.1-26 IsZeroSemigroup
      14.1-27 IsZeroSimpleSemigroup
  15 [33X[0;0YProperties and attributes specific to inverse semigroups[133X
    15.1 [33X[0;0YAttributes specific to inverse semigroups[133X
      15.1-1 NaturalLeqInverseSemigroup
      15.1-2 JoinIrreducibleDClasses
      15.1-3 MajorantClosure
      15.1-4 Minorants
      15.1-5 PrimitiveIdempotents
      15.1-6 RightCosetsOfInverseSemigroup
      15.1-7 SameMinorantsSubgroup
      15.1-8 SmallerDegreePartialPermRepresentation
      15.1-9 VagnerPrestonRepresentation
      15.1-10 CharacterTableOfInverseSemigroup
    15.2 [33X[0;0YProperties of inverse semigroups[133X
      15.2-1 IsCliffordSemigroup
      15.2-2 IsBrandtSemigroup
      15.2-3 IsEUnitaryInverseSemigroup
      15.2-4 IsFactorisableInverseMonoid
      15.2-5 IsJoinIrreducible
      15.2-6 IsMajorantlyClosed
      15.2-7 IsMonogenicInverseSemigroup
      15.2-8 IsMonogenicInverseMonoid
  16 [33X[0;0YCongruences[133X
    16.1 [33X[0;0YSemigroup congruence objects[133X
      16.1-1 IsSemigroupCongruence
      16.1-2 IsLeftSemigroupCongruence
      16.1-3 IsRightSemigroupCongruence
    16.2 [33X[0;0YCreating congruences[133X
      16.2-1 SemigroupCongruence
      16.2-2 LeftSemigroupCongruence
      16.2-3 RightSemigroupCongruence
      16.2-4 GeneratingPairsOfSemigroupCongruence
    16.3 [33X[0;0YCongruence classes[133X
      16.3-1 IsCongruenceClass
      16.3-2 IsLeftCongruenceClass
      16.3-3 IsRightCongruenceClass
      16.3-4 CongruenceClassOfElement
      16.3-5 CongruenceClasses
      16.3-6 NonTrivialEquivalenceClasses
      16.3-7 NonTrivialCongruenceClasses
      16.3-8 NrEquivalenceClasses
      16.3-9 NrCongruenceClasses
      16.3-10 EquivalenceRelationLookup
      16.3-11 EquivalenceRelationCanonicalLookup
      16.3-12 EquivalenceRelationCanonicalPartition
      16.3-13 OnLeftCongruenceClasses
      16.3-14 OnRightCongruenceClasses
    16.4 [33X[0;0YFinding the congruences of a semigroup[133X
      16.4-1 CongruencesOfSemigroup
      16.4-2 MinimalCongruencesOfSemigroup
      16.4-3 PrincipalCongruencesOfSemigroup
      16.4-4 IsCongruencePoset
      16.4-5 LatticeOfCongruences
      16.4-6 PosetOfPrincipalCongruences
      16.4-7 CongruencesOfPoset
      16.4-8 UnderlyingSemigroupOfCongruencePoset
      16.4-9 PosetOfCongruences
      16.4-10 JoinSemilatticeOfCongruences
      16.4-11 MinimalCongruences
    16.5 [33X[0;0YComparing congruences[133X
      16.5-1 IsSubrelation
      16.5-2 IsSuperrelation
      16.5-3 MeetSemigroupCongruences
      16.5-4 JoinSemigroupCongruences
    16.6 [33X[0;0YCongruences on Rees matrix semigroups[133X
      16.6-1 IsRMSCongruenceByLinkedTriple
      16.6-2 RMSCongruenceByLinkedTriple
      16.6-3 IsRMSCongruenceClassByLinkedTriple
      16.6-4 RMSCongruenceClassByLinkedTriple
      16.6-5 IsLinkedTriple
      16.6-6 CanonicalRepresentative
      16.6-7 AsSemigroupCongruenceByGeneratingPairs
      16.6-8 AsRMSCongruenceByLinkedTriple
    16.7 [33X[0;0YCongruences on inverse semigroups[133X
      16.7-1 IsInverseSemigroupCongruenceByKernelTrace
      16.7-2 InverseSemigroupCongruenceByKernelTrace
      16.7-3 AsInverseSemigroupCongruenceByKernelTrace
      16.7-4 KernelOfSemigroupCongruence
      16.7-5 TraceOfSemigroupCongruence
      16.7-6 IsInverseSemigroupCongruenceClassByKernelTrace
      16.7-7 MinimumGroupCongruence
    16.8 [33X[0;0YRees congruences[133X
      16.8-1 SemigroupIdealOfReesCongruence
      16.8-2 IsReesCongruenceClass
    16.9 [33X[0;0YUniversal congruences[133X
      16.9-1 IsUniversalSemigroupCongruence
      16.9-2 IsUniversalSemigroupCongruenceClass
      16.9-3 UniversalSemigroupCongruence
  17 [33X[0;0YSemigroup homomorphisms[133X
    17.1 [33X[0;0YIsomorphisms of arbitrary semigroups[133X
      17.1-1 IsIsomorphicSemigroup
      17.1-2 SmallestMultiplicationTable
      17.1-3 IsomorphismSemigroups
    17.2 [33X[0;0YIsomorphisms of Rees (0-)matrix semigroups[133X
      17.2-1 IsRMSIsoByTriple
      17.2-2 RMSIsoByTriple
      17.2-3 ELM_LIST
      17.2-4 CompositionMapping2
      17.2-5 ImagesElm
      17.2-6 [33X[0;0YOperators for isomorphisms of Rees (0-)matrix semigroup by
      triples[133X
  18 [33X[0;0YVisualising semigroups and elements[133X
    18.1 [33X[0;0YAutomatic viewing[133X
      18.1-1 Splash
    18.2 [33X[0;0Y[10Xdot[110X pictures[133X
      18.2-1 DotString
      18.2-2 DotSemilatticeOfIdempotents
    18.3 [33X[0;0Y[10Xtex[110X output[133X
      18.3-1 TexString
    18.4 [33X[0;0Y[10Xtikz[110X pictures[133X
      18.4-1 TikzString
  19 [33X[0;0YIO[133X
    19.1 [33X[0;0YReading and writing elements to a file[133X
      19.1-1 ReadGenerators
      19.1-2 WriteGenerators
      19.1-3 IteratorFromPickledFile
  
  
  [32X
