The MONOID Package
  
  
                                 Version 3.1.3
  
  
                                 J. D. Mitchell
  
  
  
  J. D. Mitchell
      Email:    mailto:jdm3@st-and.ac.uk
  
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  Copyright
  © 2008 J. D. Mitchell.
  
  
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  Acknowledgements
  The  author  would  like  to  thank  P.  v. Bunau, A. Distler, S. Linton, J.
  Neubueser,  V.  Maltcev, M. Neuhoeffer, M. R. Quick, E. F. Robertson, and N.
  Ruskuc  for  their  help and suggestions. Special thanks go to J. Araujo for
  his mathematical suggestions.
  
  I would also like to acknowledge the support of the Centre of Algebra at the
  University of Lisbon, and of EPSRC grant number GR/S/56085/01.
  
  
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  Colophon
  If you use the MONOID package, I would really appreciate it if you would let
  me  know  by  sending me an email to mailto:jdm3@st-and.ac.uk. If you notice
  that  there  are any features missing that you think are important or if you
  find a bug, please let me know.
  
  
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  Contents (MONOID)
  
  1 The MONOID package
    1.1 Introduction
    1.2 Installing MONOID
    1.3 Testing MONOID
    1.4 Changes
    1.5 Forthcoming Features
  2 Transformations
    2.1 Creating Transformations
      2.1-1 TransformationByKernelAndImage
      2.1-2 AllTransformationsWithKerAndImg
      2.1-3 Idempotent
      2.1-4 RandomIdempotent
      2.1-5 RandomTransformation
      2.1-6 TransformationActionNC
    2.2 Properties & Attributes
      2.2-1 IsTransversal
      2.2-2 IsKerImgOfTransformation
      2.2-3 KerImgOfTransformation
      2.2-4 IsRegularTransformation
      2.2-5 IndexPeriodOfTransformation
      2.2-6 SmallestIdempotentPower
      2.2-7 InversesOfTransformation
    2.3 Changing Representation
      2.3-1 AsBooleanMatrix
      2.3-2 AsPermOfRange
  3 Monoid Actions and Orbits
    3.1 Introduction
    3.2 Actions
      3.2-1 OnTuplesOfSetsAntiAction
      3.2-2 OnKernelsAntiAction
    3.3 General Orbits
      3.3-1 MonoidOrbit
      3.3-2 MonoidOrbits
      3.3-3 StrongOrbit
      3.3-4 StrongOrbits
      3.3-5 GradedOrbit
      3.3-6 ShortOrbit
      3.3-7 GradedStrongOrbit
      3.3-8 ShortStrongOrbit
    3.4 Some Specific Orbits
      3.4-1 ImagesOfTransSemigroup
      3.4-2 GradedImagesOfTransSemigroup
      3.4-3 KernelsOfTransSemigroup
      3.4-4 GradedKernelsOfTransSemigroup
      3.4-5 StrongOrbitOfImage
      3.4-6 StrongOrbitsOfImages
  4 Green's Relations
    4.1 Introduction
    4.2 Data Structures
      4.2-1 GreensData
      4.2-2 GreensRClassData
      4.2-3 GreensLClassData
      4.2-4 GreensHClassData
      4.2-5 GreensDClassData
      4.2-6 IsGreensData
      4.2-7 XClassData
      4.2-8 IsGreensXClassDataRep
      4.2-9 IsAssociatedSemigpTransSemigp
      4.2-10 SchutzenbergerGroup
      4.2-11 Idempotents
      4.2-12 PartialOrderOfDClasses
  5 Properties of Semigroups
    5.1 Introduction
    5.2 Property Tests
      5.2-1 IsCompletelyRegularSemigroup
      5.2-2 IsSimpleSemigroup
      5.2-3 IsGroupAsSemigroup
      5.2-4 IsCommutativeSemigroup
      5.2-5 IsRegularSemigroup
      5.2-6 IsInverseSemigroup
      5.2-7 IsCliffordSemigroup
      5.2-8 IsBand
      5.2-9 IsRectangularBand
      5.2-10 IsSemiBand
      5.2-11 IsOrthodoxSemigroup
      5.2-12 IsRightZeroSemigroup
      5.2-13 IsLeftZeroSemigroup
      5.2-14 IsZeroSemigroup
      5.2-15 IsZeroGroup
      5.2-16 MultiplicativeZero
  6 Special Classes of Semigroup
    6.1 Some Classes of Semigroup
      6.1-1 SingularSemigroup
      6.1-2 OrderPreservingSemigroup
      6.1-3 KiselmanSemigroup
    6.2 Zero Groups and Zero Semigroups
      6.2-1 ZeroSemigroup
      6.2-2 ZeroSemigroupElt
      6.2-3 ZeroGroup
      6.2-4 ZeroGroupElt
      6.2-5 UnderlyingGroupOfZG
      6.2-6 UnderlyingGroupEltOfZGElt
    6.3 Random Semigroups
      6.3-1 RandomMonoid
      6.3-2 RandomSemigroup
      6.3-3 RandomReesMatrixSemigroup
      6.3-4 RandomReesZeroMatrixSemigroup
  7 Semigroup Homomorphisms
    7.1 Introduction
      7.1-1 InfoAutos
    7.2 Creating Homomorphisms
      7.2-1 SemigroupHomomorphismByFunction
      7.2-2 SemigroupHomomorphismByImagesOfGens
      7.2-3 SemigroupHomomorphismByImages
    7.3 Inner Automorphisms
      7.3-1 InnerAutomorphismOfSemigroup
      7.3-2 ConjugatorOfInnerAutomorphismOfSemigroup
      7.3-3 IsInnerAutomorphismOfSemigroup
      7.3-4 InnerAutomorphismsOfSemigroup
      7.3-5 InnerAutomorphismsOfSemigroupInGroup
      7.3-6 InnerAutomorphismsAutomorphismGroup
      7.3-7 IsInnerAutomorphismsOfSemigroup
      7.3-8 IsInnerAutomorphismsOfZeroGroup
    7.4 Automorphism Groups
      7.4-1 AutomorphismGroup
      7.4-2 AutomorphismsSemigroupInGroup
      7.4-3 IsAutomorphismGroupOfSemigroup
      7.4-4 IsAutomorphismGroupOfSimpleSemigp
      7.4-5 IsAutomorphismGroupOfZeroGroup
      7.4-6 IsAutomorphismGroupOfZeroSemigroup
      7.4-7 IsAutomorphismGroupOfRMS
      7.4-8 IsAutomorphismGroupOfRZMS
    7.5 Rees Matrix Semigroups
      7.5-1 RMSIsoByTriple
      7.5-2 RZMSIsoByTriple
      7.5-3 IsRMSIsoByTripleRep
      7.5-4 IsRZMSIsoByTripleRep
      7.5-5 RMSInducedFunction
      7.5-6 RZMSInducedFunction
      7.5-7 RZMStoRZMSInducedFunction
      7.5-8 RZMSGraph
      7.5-9 RightTransStabAutoGroup
    7.6 Zero Groups
      7.6-1 ZeroGroupAutomorphism
      7.6-2 IsZeroGroupAutomorphismRep
      7.6-3 UnderlyingGroupAutoOfZeroGroupAuto
    7.7 Isomorphisms
      7.7-1 IsomorphismAutomorphismGroupOfRMS
      7.7-2 IsomorphismPermGroup
      7.7-3 IsomorphismFpSemigroup
      7.7-4 IsomorphismFpMonoid
      7.7-5 IsomorphismSemigroups
      7.7-6 IsomorphismReesMatrixSemigroupOfDClass
      7.7-7 IsomorphismReesMatrixSemigroup
  
  
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