  
  
                         [1XFunctionally recursive groups[0m
  
  
                              [1XSelf-similar groups[0m
  
  
                               Version 0.857142p8
  
  
                          $Date: 2008/12/02 12:04:31 $
  
  
                               Laurent Bartholdi
  
  
  
            Groups  generated  by  automata or satisfying functional
            recursions
  
  
  
  Laurent Bartholdi
      Email:    [7Xmailto:laurent.bartholdi@gmail.com[0m
      Homepage: [7Xhttp://www.uni-math.gwdg.de/laurent/[0m
  
  
  Address: Mathematisches Institut, Bunsenstraße 3-5, D-37073 Göttingen
           D-37073 Göttingen
           Germany
  
  
  -------------------------------------------------------
  [1XAbstract[0m
  This  document  describes  the package [5XFR[0m, which implements in [5XGAP[0m the basic
  objects of Mealy machines and functional recursions; and handles groups that
  they generate.
  
  Groups  defined by a recursive action on a rooted tree can be defined in [5XGAP[0m
  via  their recursion. Various algorithms are implemented to manipulate these
  groups and their elements.
  
  For  comments  or questions on [5XFR[0m please contact the author; this package is
  still under development.
  
  
  -------------------------------------------------------
  [1XCopyright[0m
  © 2006-2008 by Laurent Bartholdi
  
  
  -------------------------------------------------------
  [1XAcknowledgements[0m
  Part  of  this  work is supported by the "Swiss National Fund for Scientific
  Research"
  
  
  -------------------------------------------------------
  [1XColophon[0m
  This  project  started  in  the mid-1990s, when, as a PhD student I did many
  calculations   with   groups   generated   by  automata,  and  realized  the
  similarities  between  all  calculations; it quickly became clear that these
  calculations could be done much better by a computer than by a human.
  
  The  first routines I wrote constructed finite representations of the groups
  considered,  so  as  to  get  insight from fast calculations within [5XGAP[0m. The
  results  then  had  to  be  proved  correct  within the infinite group under
  consideration,  and  this  often  involved guessing appropriate words in the
  infinite group with a given image in the finite quotient.
  
  Around  2000,  I  had developed quite a few routines, which I assembled in a
  [5XGAP[0m  package,  that  dealt  directly  with infinite groups. This package was
  primitive at its core, but was extended with various routines as they became
  useful.
  
  I  decided  in  late  2005  to  start a new package from scratch, that would
  encorporate  as  much  functionality  as  possible in a uniform manner; that
  would  handle  semigroups  as well as groups; that could be easily extended;
  and  with  a complete, understandable documentation. I hope I am not too far
  from these objectives.
  
  
  -------------------------------------------------------
  
  
  [1XContents (FR)[0X
  
  1 Licensing
  2 FR package
    2.1 A brief mathematical introduction
    2.2 An example session
  3 Functionally recursive machines
    3.1 Types of machines
    3.2 Products of machines
    3.3 Creators for [10XFRMachine[0ms
      3.3-1 FRMachineNC
      3.3-2 FRMachine
      3.3-3 UnderlyingFRMachine
      3.3-4 AsGroupFRMachine
      3.3-5 ChangeFRMachineBasis
    3.4 Attributes for [10XFRMachine[0ms
      3.4-1 StateSet
      3.4-2 GeneratorsOfFRMachine
      3.4-3 Output
      3.4-4 Transition
      3.4-5 WreathRecursion
    3.5 Operations for [10XFRMachine[0ms
      3.5-1 StructuralGroup
      3.5-2 \+
      3.5-3 \*
      3.5-4 TensorSumOp
      3.5-5 TensorProductOp
      3.5-6 DirectSumOp
      3.5-7 DirectProductOp
      3.5-8 TreeWreathProduct
      3.5-9 SubFRMachine
      3.5-10 Minimized
      3.5-11 Correspondence
  4 Functionally recursive elements
    4.1 Creators for [10XFRElement[0ms
      4.1-1 FRElementNC
      4.1-2 FRElement
      4.1-3 FRElement
      4.1-4 ComposeElement
      4.1-5 VertexElement
      4.1-6 DiagonalElement
      4.1-7 AsGroupFRElement
    4.2 Operations and Attributes for [10XFRElement[0ms
      4.2-1 Output
      4.2-2 Activity
      4.2-3 Transition
      4.2-4 Portrait
      4.2-5 DecompositionOfFRElement
      4.2-6 StateSet
      4.2-7 State
      4.2-8 States
      4.2-9 FixedStates
      4.2-10 LimitStates
      4.2-11 IsFiniteStateFRElement
      4.2-12 InitialState
      4.2-13 \^
      4.2-14 \*
      4.2-15 \[\]
  5 Mealy machines and elements
    5.1 Creators for [10XMealyMachine[0ms and [10XMealyElement[0ms
      5.1-1 MealyMachine
      5.1-2 MealyMachine
      5.1-3 MealyMachineNC
      5.1-4 AllMealyMachines
    5.2 Operations and Attributes for [10XMealyMachine[0ms and [10XMealyElement[0ms
      5.2-1 Draw
      5.2-2 Minimized
      5.2-3 DualMachine
      5.2-4 IsReversible
      5.2-5 IsMinimized
      5.2-6 AlphabetInvolution
      5.2-7 IsBireversible
      5.2-8 StateGrowth
      5.2-9 Degree
      5.2-10 IsFinitaryFRElement
      5.2-11 Depth
      5.2-12 IsBoundedFRElement
      5.2-13 IsPolynomialGrowthFRElement
      5.2-14 Signatures
      5.2-15 VertexTransformationsFRMachine
      5.2-16 FixedRay
      5.2-17 IsLevelTransitive
      5.2-18 AsMealyMachine
      5.2-19 AsMealyMachine
      5.2-20 AsMealyElement
      5.2-21 AsIntMealyMachine
      5.2-22 TopElement
      5.2-23 ConfinalityClasses
      5.2-24 Germs
      5.2-25 HasOpenSetConditionFRElement
      5.2-26 LimitMachine
      5.2-27 NucleusMachine
      5.2-28 GuessMealyElement
  6 Linear machines and elements
    6.1 Methods and operations for [10XLinearFRMachine[0ms and [10XLinearFRElement[0ms
      6.1-1 VectorMachine
      6.1-2 AlgebraMachine
      6.1-3 Transition
      6.1-4 Transitions
      6.1-5 NestedMatrixState
      6.1-6 ActivitySparse
      6.1-7 Activities
      6.1-8 IsConvergent
      6.1-9 TransposedFRElement
      6.1-10 LDUDecompositionFRElement
      6.1-11 GuessVectorElement
      6.1-12 AsLinearMachine
      6.1-13 AsVectorMachine
      6.1-14 AsAlgebraMachine
      6.1-15 AsVectorMachine
      6.1-16 AsAlgebraMachine
  7 Self-similar groups, monoids and semigroups
    7.1 Creators for FR semigroups
      7.1-1 FRGroup
      7.1-2 SCGroup
      7.1-3 Correspondence
      7.1-4 FullSCGroup
      7.1-5 FRMachineFRGroup
      7.1-6 IsomorphismFRGroup
      7.1-7 IsomorphismMealyGroup
      7.1-8 FRGroupByVirtualEndomorphism
      7.1-9 TreeWreathProduct
      7.1-10 WeaklyBranchedEmbedding
    7.2 Operations for FR semigroups
      7.2-1 PermGroup
      7.2-2 PcGroup
      7.2-3 TransMonoid
      7.2-4 TransSemigroup
      7.2-5 EpimorphismGermGroup
      7.2-6 StabilizerImage
      7.2-7 LevelStabilizer
      7.2-8 IsStateClosedFRSemigroup
      7.2-9 StateClosure
      7.2-10 IsRecurrentFRSemigroup
      7.2-11 IsLevelTransitive
      7.2-12 IsInfinitelyTransitive
      7.2-13 IsFinitaryFRSemigroup
      7.2-14 Degree
      7.2-15 HasOpenSetConditionFRSemigroup
      7.2-16 IsContracting
      7.2-17 NucleusOfFRSemigroup
      7.2-18 NucleusMachine
      7.2-19 BranchingSubgroup
      7.2-20 FindBranchingSubgroup
      7.2-21 IsBranched
      7.2-22 IsBranchingSubgroup
      7.2-23 TopVertexTransformations
      7.2-24 VertexTransformations
      7.2-25 VirtualEndomorphism
      7.2-26 EpimorphismFromFpGroup
      7.2-27 IsomorphismSubgroupFpGroup
    7.3 Properties for infinite groups
      7.3-1 IsTorsionGroup
      7.3-2 IsTorsionFreeGroup
      7.3-3 IsAmenableGroup
      7.3-4 IsVirtuallySimpleGroup
      7.3-5 IsResiduallyFinite
      7.3-6 IsSQUniversal
      7.3-7 IsJustInfinite
  8 Algebras
    8.1 Creators for FR algebras
      8.1-1 FRAlgebra
      8.1-2 SCAlgebra
      8.1-3 BranchingIdeal
    8.2 Operations for FR algebras
      8.2-1 MatrixQuotient
      8.2-2 ThinnedAlgebra
      8.2-3 Nillity
  9 Iterated monodromy groups
    9.1 Creators and operations for IMG FR machines
      9.1-1 IMGFRMachine
      9.1-2 IMGRelator
      9.1-3 Mating
      9.1-4 PolynomialFRMachine
      9.1-5 DBRationalIMGGroup
      9.1-6 ValueRational
      9.1-7 CriticalValuesQuadraticRational
      9.1-8 CanonicalQuadraticRational
      9.1-9 PostCriticalMachine
    9.2 Spiders
      9.2-1 RationalFunction
      9.2-2 IMGFRMachine
  10 Examples
    10.1 Examples of groups
      10.1-1 FullBinaryGroup
      10.1-2 BinaryKneadingGroup
      10.1-3 BasilicaGroup
      10.1-4 AddingGroup
      10.1-5 BinaryAddingGroup
      10.1-6 MixerGroup
      10.1-7 SunicGroup
      10.1-8 GrigorchukMachines
      10.1-9 GrigorchukMachine
      10.1-10 GrigorchukOverGroup
      10.1-11 GrigorchukEvilTwin
      10.1-12 BrunnerSidkiVieiraGroup
      10.1-13 AleshinGroups
      10.1-14 AleshinGroup
      10.1-15 BabyAleshinGroup
      10.1-16 SidkiFreeGroup
      10.1-17 GuptaSidkiGroups
      10.1-18 GuptaSidkiGroup
      10.1-19 NeumannGroup
      10.1-20 FabrykowskiGuptaGroup
      10.1-21 OtherSpinalGroup
      10.1-22 GammaPQMachine
      10.1-23 HanoiGroup
      10.1-24 DahmaniGroup
      10.1-25 MamaghaniGroup
      10.1-26 WeierstrassGroup
      10.1-27 FRAffineGroup
      10.1-28 CayleyGroup
    10.2 Examples of semigroups
      10.2-1 I2Machine
      10.2-2 I4Machine
    10.3 Examples of algebras
      10.3-1 PSZAlgebra
      10.3-2 GrigorchukThinnedAlgebra
      10.3-3 GuptaSidkiThinnedAlgebra
      10.3-4 SidkiFreeAlgebra
    10.4 Bacher's determinant identities
    10.5 VH groups
      10.5-1 VHStructure
      10.5-2 VerticalAction
      10.5-3 VHGroup
      10.5-4 IsIrreducibleVHGroup
      10.5-5 MaximalSimpleSubgroup
  11 FR implementation details
    11.1 The family of FR objects
      11.1-1 FRMFamily
      11.1-2 FREFamily
      11.1-3 AlphabetOfFRObject
      11.1-4 AsPermutation
      11.1-5 AsTransformation
    11.2 Filters for [10XFRObject[0ms
      11.2-1 IsGroupFRMachine
      11.2-2 IsFRMachineStrRep
      11.2-3 IsMealyMachine
      11.2-4 IsMealyElement
      11.2-5 IsMealyMachineIntRep
      11.2-6 IsMealyMachineDomainRep
      11.2-7 IsVectorFRMachineRep
      11.2-8 IsAlgebraFRMachineRep
      11.2-9 IsLinearFRMachine
      11.2-10 IsLinearFRElement
      11.2-11 IsFRElement
      11.2-12 IsFRObject
      11.2-13 IsFRMachine
      11.2-14 IsInvertible
      11.2-15 IsFRGroup
      11.2-16 IsFRAlgebra
    11.3 Some of the algorithms implemented
      11.3-1 FRMachineRWS
      11.3-2 Order of FR elements
      11.3-3 Membership in semigroups
      11.3-4 Order of groups
      11.3-5 Images and preimages of some groups in f.p. and l.p. groups
      11.3-6 Comparison of FR, Mealy, vector, and algebra elements
      11.3-7 Inverses of linear elements
  12 Miscellanea
    12.1 Helpers
      12.1-1 maybe
      12.1-2 ReturnMaybe
      12.1-3 TensorSum
      12.1-4 TensorProductX
      12.1-5 DirectSum
      12.1-6 PeriodicList
      12.1-7 CompressPeriodicList
      12.1-8 IsConfinal
      12.1-9 ConfinalityClass
      12.1-10 LargestCommonPrefix
      12.1-11 WordGrowth
      12.1-12 ShortGroupRelations
      12.1-13 ShortGroupWordInSet
      12.1-14 SurfaceBraidFpGroup
      12.1-15 CharneyBraidFpGroup
      12.1-16 ArtinRepresentation
      12.1-17 StringByInt
      12.1-18 PositionTower
      12.1-19 CoefficientsInAbelianExtension
      12.1-20 MagmaEndomorphismByImagesNC
      12.1-21 MagmaHomomorphismByImagesNC
      12.1-22 NewFIFO
      12.1-23 ProductIdeal
      12.1-24 DimensionSeries
      12.1-25 Trans
    12.2 User settings
      12.2-1 InfoFR
      12.2-2 FR_SEARCH
  
  
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