\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Licensing}}{10}{chapter.1}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }FR package}}{11}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter }A brief mathematical introduction}}{11}{section.2.1}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }An example session}}{12}{section.2.2}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }Functionally recursive machines}}{16}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Types of machines}}{16}{section.3.1}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter }Products of machines}}{17}{section.3.2}
\contentsline {section}{\numberline {3.3}\leavevmode {\color {Chapter }Creators for \texttt {FRMachine}s}}{17}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}\leavevmode {\color {Chapter }FRMachineNC (family,free,listlist,list)}}{17}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}\leavevmode {\color {Chapter }FRMachine ([list,]list,list)}}{17}{subsection.3.3.2}
\contentsline {subsection}{\numberline {3.3.3}\leavevmode {\color {Chapter }UnderlyingFRMachine}}{18}{subsection.3.3.3}
\contentsline {subsection}{\numberline {3.3.4}\leavevmode {\color {Chapter }AsGroupFRMachine}}{19}{subsection.3.3.4}
\contentsline {subsection}{\numberline {3.3.5}\leavevmode {\color {Chapter }ChangeFRMachineBasis}}{21}{subsection.3.3.5}
\contentsline {section}{\numberline {3.4}\leavevmode {\color {Chapter }Attributes for \texttt {FRMachine}s}}{21}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}\leavevmode {\color {Chapter }StateSet (FR machine)}}{21}{subsection.3.4.1}
\contentsline {subsection}{\numberline {3.4.2}\leavevmode {\color {Chapter }GeneratorsOfFRMachine}}{22}{subsection.3.4.2}
\contentsline {subsection}{\numberline {3.4.3}\leavevmode {\color {Chapter }Output (FR machine,state)}}{22}{subsection.3.4.3}
\contentsline {subsection}{\numberline {3.4.4}\leavevmode {\color {Chapter }Transition (FR machine,state,input)}}{22}{subsection.3.4.4}
\contentsline {subsection}{\numberline {3.4.5}\leavevmode {\color {Chapter }Transitions (FR machine,state)}}{22}{subsection.3.4.5}
\contentsline {subsection}{\numberline {3.4.6}\leavevmode {\color {Chapter }WreathRecursion}}{23}{subsection.3.4.6}
\contentsline {section}{\numberline {3.5}\leavevmode {\color {Chapter }Operations for \texttt {FRMachine}s}}{23}{section.3.5}
\contentsline {subsection}{\numberline {3.5.1}\leavevmode {\color {Chapter }StructuralGroup}}{23}{subsection.3.5.1}
\contentsline {subsection}{\numberline {3.5.2}\leavevmode {\color {Chapter }\texttt {\char 92\relax }+}}{24}{subsection.3.5.2}
\contentsline {subsection}{\numberline {3.5.3}\leavevmode {\color {Chapter }\texttt {\char 92\relax }*}}{24}{subsection.3.5.3}
\contentsline {subsection}{\numberline {3.5.4}\leavevmode {\color {Chapter }TensorSumOp (FR Machines)}}{24}{subsection.3.5.4}
\contentsline {subsection}{\numberline {3.5.5}\leavevmode {\color {Chapter }TensorProductOp (FR Machines)}}{25}{subsection.3.5.5}
\contentsline {subsection}{\numberline {3.5.6}\leavevmode {\color {Chapter }DirectSumOp (FR Machines)}}{25}{subsection.3.5.6}
\contentsline {subsection}{\numberline {3.5.7}\leavevmode {\color {Chapter }DirectProductOp (FR Machines)}}{26}{subsection.3.5.7}
\contentsline {subsection}{\numberline {3.5.8}\leavevmode {\color {Chapter }TreeWreathProduct (FR machine)}}{26}{subsection.3.5.8}
\contentsline {subsection}{\numberline {3.5.9}\leavevmode {\color {Chapter }SubFRMachine}}{27}{subsection.3.5.9}
\contentsline {subsection}{\numberline {3.5.10}\leavevmode {\color {Chapter }Minimized (FR machine)}}{27}{subsection.3.5.10}
\contentsline {subsection}{\numberline {3.5.11}\leavevmode {\color {Chapter }Correspondence (FR machine)}}{27}{subsection.3.5.11}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Functionally recursive elements}}{29}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }Creators for \texttt {FRElement}s}}{29}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}\leavevmode {\color {Chapter }FRElementNC (family,free,listlist,list,assocword)}}{29}{subsection.4.1.1}
\contentsline {subsection}{\numberline {4.1.2}\leavevmode {\color {Chapter }FRElement ([list,]list,list,list)}}{30}{subsection.4.1.2}
\contentsline {subsection}{\numberline {4.1.3}\leavevmode {\color {Chapter }FRElement (machine/element,list)}}{31}{subsection.4.1.3}
\contentsline {subsection}{\numberline {4.1.4}\leavevmode {\color {Chapter }ComposeElement (elementcoll,perm)}}{31}{subsection.4.1.4}
\contentsline {subsection}{\numberline {4.1.5}\leavevmode {\color {Chapter }VertexElement}}{32}{subsection.4.1.5}
\contentsline {subsection}{\numberline {4.1.6}\leavevmode {\color {Chapter }DiagonalElement}}{32}{subsection.4.1.6}
\contentsline {subsection}{\numberline {4.1.7}\leavevmode {\color {Chapter }AsGroupFRElement}}{33}{subsection.4.1.7}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Operations and Attributes for \texttt {FRElement}s}}{34}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color {Chapter }Output (FR element)}}{34}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}\leavevmode {\color {Chapter }Activity}}{35}{subsection.4.2.2}
\contentsline {subsection}{\numberline {4.2.3}\leavevmode {\color {Chapter }Transition (FR element,input)}}{35}{subsection.4.2.3}
\contentsline {subsection}{\numberline {4.2.4}\leavevmode {\color {Chapter }Transitions (FR element)}}{36}{subsection.4.2.4}
\contentsline {subsection}{\numberline {4.2.5}\leavevmode {\color {Chapter }Portrait}}{36}{subsection.4.2.5}
\contentsline {subsection}{\numberline {4.2.6}\leavevmode {\color {Chapter }DecompositionOfFRElement}}{36}{subsection.4.2.6}
\contentsline {subsection}{\numberline {4.2.7}\leavevmode {\color {Chapter }StateSet (FR element)}}{37}{subsection.4.2.7}
\contentsline {subsection}{\numberline {4.2.8}\leavevmode {\color {Chapter }State}}{37}{subsection.4.2.8}
\contentsline {subsection}{\numberline {4.2.9}\leavevmode {\color {Chapter }States}}{37}{subsection.4.2.9}
\contentsline {subsection}{\numberline {4.2.10}\leavevmode {\color {Chapter }FixedStates}}{38}{subsection.4.2.10}
\contentsline {subsection}{\numberline {4.2.11}\leavevmode {\color {Chapter }LimitStates}}{38}{subsection.4.2.11}
\contentsline {subsection}{\numberline {4.2.12}\leavevmode {\color {Chapter }IsFiniteStateFRElement}}{39}{subsection.4.2.12}
\contentsline {subsection}{\numberline {4.2.13}\leavevmode {\color {Chapter }NucleusOfFRMachine}}{39}{subsection.4.2.13}
\contentsline {subsection}{\numberline {4.2.14}\leavevmode {\color {Chapter }InitialState}}{40}{subsection.4.2.14}
\contentsline {subsection}{\numberline {4.2.15}\leavevmode {\color {Chapter }\texttt {\char 92\relax }\texttt {\char 94\relax } (POW)}}{40}{subsection.4.2.15}
\contentsline {subsection}{\numberline {4.2.16}\leavevmode {\color {Chapter }\texttt {\char 92\relax }* (PROD)}}{40}{subsection.4.2.16}
\contentsline {subsection}{\numberline {4.2.17}\leavevmode {\color {Chapter }\texttt {\char 92\relax }[\texttt {\char 92\relax }] (ELMLIST)}}{41}{subsection.4.2.17}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }Mealy machines and elements}}{42}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter }Creators for \texttt {MealyMachine}s and \texttt {MealyElement}s}}{42}{section.5.1}
\contentsline {subsection}{\numberline {5.1.1}\leavevmode {\color {Chapter }MealyMachine ([list,]listlist,list)}}{42}{subsection.5.1.1}
\contentsline {subsection}{\numberline {5.1.2}\leavevmode {\color {Chapter }MealyMachine (domain,domain,function,function)}}{43}{subsection.5.1.2}
\contentsline {subsection}{\numberline {5.1.3}\leavevmode {\color {Chapter }MealyMachineNC (family,listlist,list)}}{44}{subsection.5.1.3}
\contentsline {subsection}{\numberline {5.1.4}\leavevmode {\color {Chapter }AllMealyMachines}}{44}{subsection.5.1.4}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter }Operations and Attributes for \texttt {MealyMachine}s and \texttt {MealyElement}s}}{45}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color {Chapter }Draw}}{45}{subsection.5.2.1}
\contentsline {subsection}{\numberline {5.2.2}\leavevmode {\color {Chapter }Minimized (Mealy machine)}}{46}{subsection.5.2.2}
\contentsline {subsection}{\numberline {5.2.3}\leavevmode {\color {Chapter }DualMachine}}{47}{subsection.5.2.3}
\contentsline {subsection}{\numberline {5.2.4}\leavevmode {\color {Chapter }IsReversible}}{47}{subsection.5.2.4}
\contentsline {subsection}{\numberline {5.2.5}\leavevmode {\color {Chapter }IsMinimized}}{47}{subsection.5.2.5}
\contentsline {subsection}{\numberline {5.2.6}\leavevmode {\color {Chapter }AlphabetInvolution}}{48}{subsection.5.2.6}
\contentsline {subsection}{\numberline {5.2.7}\leavevmode {\color {Chapter }IsBireversible}}{48}{subsection.5.2.7}
\contentsline {subsection}{\numberline {5.2.8}\leavevmode {\color {Chapter }StateGrowth}}{48}{subsection.5.2.8}
\contentsline {subsection}{\numberline {5.2.9}\leavevmode {\color {Chapter }Degree (FR element)}}{48}{subsection.5.2.9}
\contentsline {subsection}{\numberline {5.2.10}\leavevmode {\color {Chapter }IsFinitaryFRElement}}{49}{subsection.5.2.10}
\contentsline {subsection}{\numberline {5.2.11}\leavevmode {\color {Chapter }Depth (FR element)}}{49}{subsection.5.2.11}
\contentsline {subsection}{\numberline {5.2.12}\leavevmode {\color {Chapter }IsBoundedFRElement}}{50}{subsection.5.2.12}
\contentsline {subsection}{\numberline {5.2.13}\leavevmode {\color {Chapter }IsPolynomialGrowthFRElement}}{50}{subsection.5.2.13}
\contentsline {subsection}{\numberline {5.2.14}\leavevmode {\color {Chapter }Signatures}}{51}{subsection.5.2.14}
\contentsline {subsection}{\numberline {5.2.15}\leavevmode {\color {Chapter }VertexTransformationsFRMachine}}{51}{subsection.5.2.15}
\contentsline {subsection}{\numberline {5.2.16}\leavevmode {\color {Chapter }FixedRay (FR element)}}{51}{subsection.5.2.16}
\contentsline {subsection}{\numberline {5.2.17}\leavevmode {\color {Chapter }IsLevelTransitive (FR element)}}{52}{subsection.5.2.17}
\contentsline {subsection}{\numberline {5.2.18}\leavevmode {\color {Chapter }AsMealyMachine (FR machine)}}{52}{subsection.5.2.18}
\contentsline {subsection}{\numberline {5.2.19}\leavevmode {\color {Chapter }AsMealyMachine (List)}}{53}{subsection.5.2.19}
\contentsline {subsection}{\numberline {5.2.20}\leavevmode {\color {Chapter }AsMealyElement}}{53}{subsection.5.2.20}
\contentsline {subsection}{\numberline {5.2.21}\leavevmode {\color {Chapter }AsIntMealyMachine}}{54}{subsection.5.2.21}
\contentsline {subsection}{\numberline {5.2.22}\leavevmode {\color {Chapter }TopElement}}{54}{subsection.5.2.22}
\contentsline {subsection}{\numberline {5.2.23}\leavevmode {\color {Chapter }ConfinalityClasses}}{54}{subsection.5.2.23}
\contentsline {subsection}{\numberline {5.2.24}\leavevmode {\color {Chapter }Germs}}{55}{subsection.5.2.24}
\contentsline {subsection}{\numberline {5.2.25}\leavevmode {\color {Chapter }HasOpenSetConditionFRElement}}{56}{subsection.5.2.25}
\contentsline {subsection}{\numberline {5.2.26}\leavevmode {\color {Chapter }LimitMachine}}{56}{subsection.5.2.26}
\contentsline {subsection}{\numberline {5.2.27}\leavevmode {\color {Chapter }NucleusMachine (FR machine)}}{56}{subsection.5.2.27}
\contentsline {subsection}{\numberline {5.2.28}\leavevmode {\color {Chapter }GuessMealyElement}}{57}{subsection.5.2.28}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }Linear machines and elements}}{58}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter }Methods and operations for \texttt {LinearFRMachine}s and \texttt {LinearFRElement}s}}{58}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}\leavevmode {\color {Chapter }VectorMachine}}{58}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}\leavevmode {\color {Chapter }AlgebraMachine}}{59}{subsection.6.1.2}
\contentsline {subsection}{\numberline {6.1.3}\leavevmode {\color {Chapter }Transition (Linear machine)}}{60}{subsection.6.1.3}
\contentsline {subsection}{\numberline {6.1.4}\leavevmode {\color {Chapter }Transitions}}{60}{subsection.6.1.4}
\contentsline {subsection}{\numberline {6.1.5}\leavevmode {\color {Chapter }NestedMatrixState}}{61}{subsection.6.1.5}
\contentsline {subsection}{\numberline {6.1.6}\leavevmode {\color {Chapter }ActivitySparse}}{61}{subsection.6.1.6}
\contentsline {subsection}{\numberline {6.1.7}\leavevmode {\color {Chapter }Activities}}{62}{subsection.6.1.7}
\contentsline {subsection}{\numberline {6.1.8}\leavevmode {\color {Chapter }IsConvergent}}{62}{subsection.6.1.8}
\contentsline {subsection}{\numberline {6.1.9}\leavevmode {\color {Chapter }TransposedFRElement}}{62}{subsection.6.1.9}
\contentsline {subsection}{\numberline {6.1.10}\leavevmode {\color {Chapter }LDUDecompositionFRElement}}{63}{subsection.6.1.10}
\contentsline {subsection}{\numberline {6.1.11}\leavevmode {\color {Chapter }GuessVectorElement}}{64}{subsection.6.1.11}
\contentsline {subsection}{\numberline {6.1.12}\leavevmode {\color {Chapter }AsLinearMachine}}{64}{subsection.6.1.12}
\contentsline {subsection}{\numberline {6.1.13}\leavevmode {\color {Chapter }AsVectorMachine}}{65}{subsection.6.1.13}
\contentsline {subsection}{\numberline {6.1.14}\leavevmode {\color {Chapter }AsAlgebraMachine}}{65}{subsection.6.1.14}
\contentsline {subsection}{\numberline {6.1.15}\leavevmode {\color {Chapter }AsVectorMachine (Linear machine)}}{66}{subsection.6.1.15}
\contentsline {subsection}{\numberline {6.1.16}\leavevmode {\color {Chapter }AsAlgebraMachine (Linear machine)}}{66}{subsection.6.1.16}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter }Self-similar groups, monoids and semigroups}}{67}{chapter.7}
\contentsline {section}{\numberline {7.1}\leavevmode {\color {Chapter }Creators for FR semigroups}}{67}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}\leavevmode {\color {Chapter }FRGroup}}{67}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}\leavevmode {\color {Chapter }SCGroup}}{69}{subsection.7.1.2}
\contentsline {subsection}{\numberline {7.1.3}\leavevmode {\color {Chapter }Correspondence (FR semigroup)}}{70}{subsection.7.1.3}
\contentsline {subsection}{\numberline {7.1.4}\leavevmode {\color {Chapter }FullSCGroup}}{70}{subsection.7.1.4}
\contentsline {subsection}{\numberline {7.1.5}\leavevmode {\color {Chapter }FRMachineFRGroup}}{71}{subsection.7.1.5}
\contentsline {subsection}{\numberline {7.1.6}\leavevmode {\color {Chapter }IsomorphismFRGroup}}{72}{subsection.7.1.6}
\contentsline {subsection}{\numberline {7.1.7}\leavevmode {\color {Chapter }IsomorphismMealyGroup}}{73}{subsection.7.1.7}
\contentsline {subsection}{\numberline {7.1.8}\leavevmode {\color {Chapter }FRGroupByVirtualEndomorphism}}{74}{subsection.7.1.8}
\contentsline {subsection}{\numberline {7.1.9}\leavevmode {\color {Chapter }TreeWreathProduct (FR group)}}{75}{subsection.7.1.9}
\contentsline {subsection}{\numberline {7.1.10}\leavevmode {\color {Chapter }WeaklyBranchedEmbedding}}{75}{subsection.7.1.10}
\contentsline {section}{\numberline {7.2}\leavevmode {\color {Chapter }Operations for FR semigroups}}{75}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}\leavevmode {\color {Chapter }PermGroup}}{75}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}\leavevmode {\color {Chapter }PcGroup}}{76}{subsection.7.2.2}
\contentsline {subsection}{\numberline {7.2.3}\leavevmode {\color {Chapter }TransMonoid}}{77}{subsection.7.2.3}
\contentsline {subsection}{\numberline {7.2.4}\leavevmode {\color {Chapter }TransSemigroup}}{77}{subsection.7.2.4}
\contentsline {subsection}{\numberline {7.2.5}\leavevmode {\color {Chapter }EpimorphismGermGroup}}{78}{subsection.7.2.5}
\contentsline {subsection}{\numberline {7.2.6}\leavevmode {\color {Chapter }StabilizerImage}}{78}{subsection.7.2.6}
\contentsline {subsection}{\numberline {7.2.7}\leavevmode {\color {Chapter }LevelStabilizer}}{79}{subsection.7.2.7}
\contentsline {subsection}{\numberline {7.2.8}\leavevmode {\color {Chapter }IsStateClosedFRSemigroup}}{79}{subsection.7.2.8}
\contentsline {subsection}{\numberline {7.2.9}\leavevmode {\color {Chapter }StateClosure}}{80}{subsection.7.2.9}
\contentsline {subsection}{\numberline {7.2.10}\leavevmode {\color {Chapter }IsRecurrentFRSemigroup}}{80}{subsection.7.2.10}
\contentsline {subsection}{\numberline {7.2.11}\leavevmode {\color {Chapter }IsLevelTransitive (FR group)}}{80}{subsection.7.2.11}
\contentsline {subsection}{\numberline {7.2.12}\leavevmode {\color {Chapter }IsInfinitelyTransitive}}{81}{subsection.7.2.12}
\contentsline {subsection}{\numberline {7.2.13}\leavevmode {\color {Chapter }IsFinitaryFRSemigroup}}{81}{subsection.7.2.13}
\contentsline {subsection}{\numberline {7.2.14}\leavevmode {\color {Chapter }Degree (FR semigroup)}}{82}{subsection.7.2.14}
\contentsline {subsection}{\numberline {7.2.15}\leavevmode {\color {Chapter }HasOpenSetConditionFRSemigroup}}{82}{subsection.7.2.15}
\contentsline {subsection}{\numberline {7.2.16}\leavevmode {\color {Chapter }IsContracting}}{82}{subsection.7.2.16}
\contentsline {subsection}{\numberline {7.2.17}\leavevmode {\color {Chapter }NucleusOfFRSemigroup}}{83}{subsection.7.2.17}
\contentsline {subsection}{\numberline {7.2.18}\leavevmode {\color {Chapter }NucleusMachine (FR semigroup)}}{83}{subsection.7.2.18}
\contentsline {subsection}{\numberline {7.2.19}\leavevmode {\color {Chapter }AdjacencyBasesWithOne}}{83}{subsection.7.2.19}
\contentsline {subsection}{\numberline {7.2.20}\leavevmode {\color {Chapter }BranchingSubgroup}}{84}{subsection.7.2.20}
\contentsline {subsection}{\numberline {7.2.21}\leavevmode {\color {Chapter }FindBranchingSubgroup}}{85}{subsection.7.2.21}
\contentsline {subsection}{\numberline {7.2.22}\leavevmode {\color {Chapter }IsBranched (FR group)}}{85}{subsection.7.2.22}
\contentsline {subsection}{\numberline {7.2.23}\leavevmode {\color {Chapter }IsBranchingSubgroup (FR semigroup)}}{85}{subsection.7.2.23}
\contentsline {subsection}{\numberline {7.2.24}\leavevmode {\color {Chapter }TopVertexTransformations}}{86}{subsection.7.2.24}
\contentsline {subsection}{\numberline {7.2.25}\leavevmode {\color {Chapter }VertexTransformations (FR semigroup)}}{86}{subsection.7.2.25}
\contentsline {subsection}{\numberline {7.2.26}\leavevmode {\color {Chapter }VirtualEndomorphism}}{86}{subsection.7.2.26}
\contentsline {subsection}{\numberline {7.2.27}\leavevmode {\color {Chapter }EpimorphismFromFpGroup}}{87}{subsection.7.2.27}
\contentsline {subsection}{\numberline {7.2.28}\leavevmode {\color {Chapter }IsomorphismSubgroupFpGroup}}{87}{subsection.7.2.28}
\contentsline {section}{\numberline {7.3}\leavevmode {\color {Chapter }Properties for infinite groups}}{88}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}\leavevmode {\color {Chapter }IsTorsionGroup}}{88}{subsection.7.3.1}
\contentsline {subsection}{\numberline {7.3.2}\leavevmode {\color {Chapter }IsTorsionFreeGroup}}{88}{subsection.7.3.2}
\contentsline {subsection}{\numberline {7.3.3}\leavevmode {\color {Chapter }IsAmenableGroup}}{89}{subsection.7.3.3}
\contentsline {subsection}{\numberline {7.3.4}\leavevmode {\color {Chapter }IsVirtuallySimpleGroup}}{89}{subsection.7.3.4}
\contentsline {subsection}{\numberline {7.3.5}\leavevmode {\color {Chapter }IsResiduallyFinite}}{89}{subsection.7.3.5}
\contentsline {subsection}{\numberline {7.3.6}\leavevmode {\color {Chapter }IsSQUniversal}}{90}{subsection.7.3.6}
\contentsline {subsection}{\numberline {7.3.7}\leavevmode {\color {Chapter }IsJustInfinite}}{90}{subsection.7.3.7}
\contentsline {chapter}{\numberline {8}\leavevmode {\color {Chapter }Algebras}}{91}{chapter.8}
\contentsline {section}{\numberline {8.1}\leavevmode {\color {Chapter }Creators for FR algebras}}{91}{section.8.1}
\contentsline {subsection}{\numberline {8.1.1}\leavevmode {\color {Chapter }FRAlgebra}}{91}{subsection.8.1.1}
\contentsline {subsection}{\numberline {8.1.2}\leavevmode {\color {Chapter }SCAlgebra}}{92}{subsection.8.1.2}
\contentsline {subsection}{\numberline {8.1.3}\leavevmode {\color {Chapter }BranchingIdeal}}{92}{subsection.8.1.3}
\contentsline {section}{\numberline {8.2}\leavevmode {\color {Chapter }Operations for FR algebras}}{92}{section.8.2}
\contentsline {subsection}{\numberline {8.2.1}\leavevmode {\color {Chapter }MatrixQuotient}}{92}{subsection.8.2.1}
\contentsline {subsection}{\numberline {8.2.2}\leavevmode {\color {Chapter }ThinnedAlgebra}}{93}{subsection.8.2.2}
\contentsline {subsection}{\numberline {8.2.3}\leavevmode {\color {Chapter }Nillity}}{93}{subsection.8.2.3}
\contentsline {subsection}{\numberline {8.2.4}\leavevmode {\color {Chapter }DegreeOfHomogeneousElement}}{93}{subsection.8.2.4}
\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter }Iterated monodromy groups}}{94}{chapter.9}
\contentsline {section}{\numberline {9.1}\leavevmode {\color {Chapter }Creators and operations for IMG machines}}{94}{section.9.1}
\contentsline {subsection}{\numberline {9.1.1}\leavevmode {\color {Chapter }AsIMGMachine (FR machine)}}{94}{subsection.9.1.1}
\contentsline {subsection}{\numberline {9.1.2}\leavevmode {\color {Chapter }IMGRelator}}{95}{subsection.9.1.2}
\contentsline {subsection}{\numberline {9.1.3}\leavevmode {\color {Chapter }IsKneadingMachine}}{95}{subsection.9.1.3}
\contentsline {subsection}{\numberline {9.1.4}\leavevmode {\color {Chapter }AsPolynomialFRMachine}}{96}{subsection.9.1.4}
\contentsline {subsection}{\numberline {9.1.5}\leavevmode {\color {Chapter }AddingElement (FR machine)}}{97}{subsection.9.1.5}
\contentsline {subsection}{\numberline {9.1.6}\leavevmode {\color {Chapter }PolynomialFRMachine}}{97}{subsection.9.1.6}
\contentsline {subsection}{\numberline {9.1.7}\leavevmode {\color {Chapter }ExternalAngles}}{99}{subsection.9.1.7}
\contentsline {subsection}{\numberline {9.1.8}\leavevmode {\color {Chapter }Lift}}{99}{subsection.9.1.8}
\contentsline {subsection}{\numberline {9.1.9}\leavevmode {\color {Chapter }AsGroupFRMachine (endomorphism)}}{100}{subsection.9.1.9}
\contentsline {subsection}{\numberline {9.1.10}\leavevmode {\color {Chapter }NormalizedPolynomialFRMachine}}{101}{subsection.9.1.10}
\contentsline {subsection}{\numberline {9.1.11}\leavevmode {\color {Chapter }Mating}}{101}{subsection.9.1.11}
\contentsline {subsection}{\numberline {9.1.12}\leavevmode {\color {Chapter }DBRationalIMGGroup}}{102}{subsection.9.1.12}
\contentsline {subsection}{\numberline {9.1.13}\leavevmode {\color {Chapter }PostCriticalMachine}}{102}{subsection.9.1.13}
\contentsline {section}{\numberline {9.2}\leavevmode {\color {Chapter }Spiders}}{103}{section.9.2}
\contentsline {subsection}{\numberline {9.2.1}\leavevmode {\color {Chapter }DelaunayTriangulation}}{103}{subsection.9.2.1}
\contentsline {subsection}{\numberline {9.2.2}\leavevmode {\color {Chapter }LocateInTriangulation}}{104}{subsection.9.2.2}
\contentsline {subsection}{\numberline {9.2.3}\leavevmode {\color {Chapter }RationalFunction}}{104}{subsection.9.2.3}
\contentsline {subsection}{\numberline {9.2.4}\leavevmode {\color {Chapter }FRMachine (rational function)}}{104}{subsection.9.2.4}
\contentsline {subsection}{\numberline {9.2.5}\leavevmode {\color {Chapter }FindThurstonObstruction}}{105}{subsection.9.2.5}
\contentsline {chapter}{\numberline {10}\leavevmode {\color {Chapter }Examples}}{106}{chapter.10}
\contentsline {section}{\numberline {10.1}\leavevmode {\color {Chapter }Examples of groups}}{106}{section.10.1}
\contentsline {subsection}{\numberline {10.1.1}\leavevmode {\color {Chapter }FullBinaryGroup}}{106}{subsection.10.1.1}
\contentsline {subsection}{\numberline {10.1.2}\leavevmode {\color {Chapter }BinaryKneadingGroup}}{106}{subsection.10.1.2}
\contentsline {subsection}{\numberline {10.1.3}\leavevmode {\color {Chapter }BasilicaGroup}}{107}{subsection.10.1.3}
\contentsline {subsection}{\numberline {10.1.4}\leavevmode {\color {Chapter }FornaessSibonyGroup}}{107}{subsection.10.1.4}
\contentsline {subsection}{\numberline {10.1.5}\leavevmode {\color {Chapter }AddingGroup}}{108}{subsection.10.1.5}
\contentsline {subsection}{\numberline {10.1.6}\leavevmode {\color {Chapter }BinaryAddingGroup}}{108}{subsection.10.1.6}
\contentsline {subsection}{\numberline {10.1.7}\leavevmode {\color {Chapter }MixerGroup}}{109}{subsection.10.1.7}
\contentsline {subsection}{\numberline {10.1.8}\leavevmode {\color {Chapter }SunicGroup}}{109}{subsection.10.1.8}
\contentsline {subsection}{\numberline {10.1.9}\leavevmode {\color {Chapter }GrigorchukMachines}}{110}{subsection.10.1.9}
\contentsline {subsection}{\numberline {10.1.10}\leavevmode {\color {Chapter }GrigorchukMachine}}{110}{subsection.10.1.10}
\contentsline {subsection}{\numberline {10.1.11}\leavevmode {\color {Chapter }GrigorchukOverGroup}}{110}{subsection.10.1.11}
\contentsline {subsection}{\numberline {10.1.12}\leavevmode {\color {Chapter }GrigorchukTwistedTwin}}{111}{subsection.10.1.12}
\contentsline {subsection}{\numberline {10.1.13}\leavevmode {\color {Chapter }BrunnerSidkiVieiraGroup}}{111}{subsection.10.1.13}
\contentsline {subsection}{\numberline {10.1.14}\leavevmode {\color {Chapter }AleshinGroups}}{112}{subsection.10.1.14}
\contentsline {subsection}{\numberline {10.1.15}\leavevmode {\color {Chapter }AleshinGroup}}{112}{subsection.10.1.15}
\contentsline {subsection}{\numberline {10.1.16}\leavevmode {\color {Chapter }BabyAleshinGroup}}{112}{subsection.10.1.16}
\contentsline {subsection}{\numberline {10.1.17}\leavevmode {\color {Chapter }SidkiFreeGroup}}{113}{subsection.10.1.17}
\contentsline {subsection}{\numberline {10.1.18}\leavevmode {\color {Chapter }GuptaSidkiGroups}}{113}{subsection.10.1.18}
\contentsline {subsection}{\numberline {10.1.19}\leavevmode {\color {Chapter }GuptaSidkiGroup}}{113}{subsection.10.1.19}
\contentsline {subsection}{\numberline {10.1.20}\leavevmode {\color {Chapter }NeumannGroup}}{113}{subsection.10.1.20}
\contentsline {subsection}{\numberline {10.1.21}\leavevmode {\color {Chapter }FabrykowskiGuptaGroup}}{114}{subsection.10.1.21}
\contentsline {subsection}{\numberline {10.1.22}\leavevmode {\color {Chapter }ZugadiSpinalGroup}}{114}{subsection.10.1.22}
\contentsline {subsection}{\numberline {10.1.23}\leavevmode {\color {Chapter }GammaPQMachine}}{114}{subsection.10.1.23}
\contentsline {subsection}{\numberline {10.1.24}\leavevmode {\color {Chapter }RattaggiGroup}}{114}{subsection.10.1.24}
\contentsline {subsection}{\numberline {10.1.25}\leavevmode {\color {Chapter }HanoiGroup}}{115}{subsection.10.1.25}
\contentsline {subsection}{\numberline {10.1.26}\leavevmode {\color {Chapter }DahmaniGroup}}{115}{subsection.10.1.26}
\contentsline {subsection}{\numberline {10.1.27}\leavevmode {\color {Chapter }MamaghaniGroup}}{116}{subsection.10.1.27}
\contentsline {subsection}{\numberline {10.1.28}\leavevmode {\color {Chapter }WeierstrassGroup}}{116}{subsection.10.1.28}
\contentsline {subsection}{\numberline {10.1.29}\leavevmode {\color {Chapter }FRAffineGroup}}{116}{subsection.10.1.29}
\contentsline {subsection}{\numberline {10.1.30}\leavevmode {\color {Chapter }CayleyGroup}}{117}{subsection.10.1.30}
\contentsline {section}{\numberline {10.2}\leavevmode {\color {Chapter }Examples of semigroups}}{118}{section.10.2}
\contentsline {subsection}{\numberline {10.2.1}\leavevmode {\color {Chapter }I2Machine}}{118}{subsection.10.2.1}
\contentsline {subsection}{\numberline {10.2.2}\leavevmode {\color {Chapter }I4Machine}}{118}{subsection.10.2.2}
\contentsline {section}{\numberline {10.3}\leavevmode {\color {Chapter }Examples of algebras}}{118}{section.10.3}
\contentsline {subsection}{\numberline {10.3.1}\leavevmode {\color {Chapter }PSZAlgebra}}{118}{subsection.10.3.1}
\contentsline {subsection}{\numberline {10.3.2}\leavevmode {\color {Chapter }GrigorchukThinnedAlgebra}}{119}{subsection.10.3.2}
\contentsline {subsection}{\numberline {10.3.3}\leavevmode {\color {Chapter }GuptaSidkiThinnedAlgebra}}{119}{subsection.10.3.3}
\contentsline {subsection}{\numberline {10.3.4}\leavevmode {\color {Chapter }GrigorchukLieAlgebra}}{120}{subsection.10.3.4}
\contentsline {subsection}{\numberline {10.3.5}\leavevmode {\color {Chapter }SidkiFreeAlgebra}}{120}{subsection.10.3.5}
\contentsline {subsection}{\numberline {10.3.6}\leavevmode {\color {Chapter }SidkiMonomialAlgebra}}{120}{subsection.10.3.6}
\contentsline {section}{\numberline {10.4}\leavevmode {\color {Chapter }Bacher's determinant identities}}{121}{section.10.4}
\contentsline {section}{\numberline {10.5}\leavevmode {\color {Chapter }VH groups}}{124}{section.10.5}
\contentsline {subsection}{\numberline {10.5.1}\leavevmode {\color {Chapter }VHStructure}}{125}{subsection.10.5.1}
\contentsline {subsection}{\numberline {10.5.2}\leavevmode {\color {Chapter }VerticalAction}}{125}{subsection.10.5.2}
\contentsline {subsection}{\numberline {10.5.3}\leavevmode {\color {Chapter }VHGroup}}{126}{subsection.10.5.3}
\contentsline {subsection}{\numberline {10.5.4}\leavevmode {\color {Chapter }IsIrreducibleVHGroup}}{126}{subsection.10.5.4}
\contentsline {subsection}{\numberline {10.5.5}\leavevmode {\color {Chapter }MaximalSimpleSubgroup}}{126}{subsection.10.5.5}
\contentsline {chapter}{\numberline {11}\leavevmode {\color {Chapter }FR implementation details}}{128}{chapter.11}
\contentsline {section}{\numberline {11.1}\leavevmode {\color {Chapter }The family of FR objects}}{128}{section.11.1}
\contentsline {subsection}{\numberline {11.1.1}\leavevmode {\color {Chapter }FRMFamily}}{128}{subsection.11.1.1}
\contentsline {subsection}{\numberline {11.1.2}\leavevmode {\color {Chapter }FREFamily}}{128}{subsection.11.1.2}
\contentsline {subsection}{\numberline {11.1.3}\leavevmode {\color {Chapter }AlphabetOfFRObject}}{128}{subsection.11.1.3}
\contentsline {subsection}{\numberline {11.1.4}\leavevmode {\color {Chapter }AsPermutation (FR object)}}{129}{subsection.11.1.4}
\contentsline {subsection}{\numberline {11.1.5}\leavevmode {\color {Chapter }AsTransformation (FR object)}}{129}{subsection.11.1.5}
\contentsline {section}{\numberline {11.2}\leavevmode {\color {Chapter }Filters for \texttt {FRObject}s}}{129}{section.11.2}
\contentsline {subsection}{\numberline {11.2.1}\leavevmode {\color {Chapter }IsGroupFRMachine}}{129}{subsection.11.2.1}
\contentsline {subsection}{\numberline {11.2.2}\leavevmode {\color {Chapter }IsFRMachineStrRep}}{129}{subsection.11.2.2}
\contentsline {subsection}{\numberline {11.2.3}\leavevmode {\color {Chapter }IsMealyMachine}}{129}{subsection.11.2.3}
\contentsline {subsection}{\numberline {11.2.4}\leavevmode {\color {Chapter }IsMealyElement}}{130}{subsection.11.2.4}
\contentsline {subsection}{\numberline {11.2.5}\leavevmode {\color {Chapter }IsMealyMachineIntRep}}{130}{subsection.11.2.5}
\contentsline {subsection}{\numberline {11.2.6}\leavevmode {\color {Chapter }IsMealyMachineDomainRep}}{130}{subsection.11.2.6}
\contentsline {subsection}{\numberline {11.2.7}\leavevmode {\color {Chapter }IsVectorFRMachineRep}}{130}{subsection.11.2.7}
\contentsline {subsection}{\numberline {11.2.8}\leavevmode {\color {Chapter }IsAlgebraFRMachineRep}}{130}{subsection.11.2.8}
\contentsline {subsection}{\numberline {11.2.9}\leavevmode {\color {Chapter }IsLinearFRMachine}}{130}{subsection.11.2.9}
\contentsline {subsection}{\numberline {11.2.10}\leavevmode {\color {Chapter }IsLinearFRElement}}{131}{subsection.11.2.10}
\contentsline {subsection}{\numberline {11.2.11}\leavevmode {\color {Chapter }IsFRElement}}{131}{subsection.11.2.11}
\contentsline {subsection}{\numberline {11.2.12}\leavevmode {\color {Chapter }IsFRObject}}{131}{subsection.11.2.12}
\contentsline {subsection}{\numberline {11.2.13}\leavevmode {\color {Chapter }IsFRMachine}}{131}{subsection.11.2.13}
\contentsline {subsection}{\numberline {11.2.14}\leavevmode {\color {Chapter }IsInvertible}}{131}{subsection.11.2.14}
\contentsline {subsection}{\numberline {11.2.15}\leavevmode {\color {Chapter }IsFRGroup}}{132}{subsection.11.2.15}
\contentsline {subsection}{\numberline {11.2.16}\leavevmode {\color {Chapter }IsFRAlgebra}}{132}{subsection.11.2.16}
\contentsline {section}{\numberline {11.3}\leavevmode {\color {Chapter }Some of the algorithms implemented}}{132}{section.11.3}
\contentsline {subsection}{\numberline {11.3.1}\leavevmode {\color {Chapter }FRMachineRWS}}{132}{subsection.11.3.1}
\contentsline {subsection}{\numberline {11.3.2}\leavevmode {\color {Chapter }Order of FR elements}}{132}{subsection.11.3.2}
\contentsline {subsection}{\numberline {11.3.3}\leavevmode {\color {Chapter }Membership in semigroups}}{133}{subsection.11.3.3}
\contentsline {subsection}{\numberline {11.3.4}\leavevmode {\color {Chapter }Order of groups}}{133}{subsection.11.3.4}
\contentsline {subsection}{\numberline {11.3.5}\leavevmode {\color {Chapter }Images and preimages of some groups in f.p. and l.p. groups}}{133}{subsection.11.3.5}
\contentsline {subsection}{\numberline {11.3.6}\leavevmode {\color {Chapter }Comparison of FR, Mealy, vector, and algebra elements}}{134}{subsection.11.3.6}
\contentsline {subsection}{\numberline {11.3.7}\leavevmode {\color {Chapter }Inverses of linear elements}}{134}{subsection.11.3.7}
\contentsline {subsection}{\numberline {11.3.8}\leavevmode {\color {Chapter }Spiders}}{135}{subsection.11.3.8}
\contentsline {chapter}{\numberline {12}\leavevmode {\color {Chapter }Miscellanea}}{137}{chapter.12}
\contentsline {section}{\numberline {12.1}\leavevmode {\color {Chapter }Helpers}}{137}{section.12.1}
\contentsline {subsection}{\numberline {12.1.1}\leavevmode {\color {Chapter }TensorSum}}{137}{subsection.12.1.1}
\contentsline {subsection}{\numberline {12.1.2}\leavevmode {\color {Chapter }TensorProduct}}{137}{subsection.12.1.2}
\contentsline {subsection}{\numberline {12.1.3}\leavevmode {\color {Chapter }DirectSum}}{137}{subsection.12.1.3}
\contentsline {subsection}{\numberline {12.1.4}\leavevmode {\color {Chapter }PeriodicList}}{137}{subsection.12.1.4}
\contentsline {subsection}{\numberline {12.1.5}\leavevmode {\color {Chapter }CompressPeriodicList}}{138}{subsection.12.1.5}
\contentsline {subsection}{\numberline {12.1.6}\leavevmode {\color {Chapter }IsConfinal}}{138}{subsection.12.1.6}
\contentsline {subsection}{\numberline {12.1.7}\leavevmode {\color {Chapter }ConfinalityClass}}{139}{subsection.12.1.7}
\contentsline {subsection}{\numberline {12.1.8}\leavevmode {\color {Chapter }LargestCommonPrefix}}{139}{subsection.12.1.8}
\contentsline {subsection}{\numberline {12.1.9}\leavevmode {\color {Chapter }WordGrowth}}{139}{subsection.12.1.9}
\contentsline {subsection}{\numberline {12.1.10}\leavevmode {\color {Chapter }ShortGroupRelations}}{141}{subsection.12.1.10}
\contentsline {subsection}{\numberline {12.1.11}\leavevmode {\color {Chapter }ShortGroupWordInSet}}{142}{subsection.12.1.11}
\contentsline {subsection}{\numberline {12.1.12}\leavevmode {\color {Chapter }SurfaceBraidFpGroup}}{142}{subsection.12.1.12}
\contentsline {subsection}{\numberline {12.1.13}\leavevmode {\color {Chapter }CharneyBraidFpGroup}}{142}{subsection.12.1.13}
\contentsline {subsection}{\numberline {12.1.14}\leavevmode {\color {Chapter }ArtinRepresentation}}{143}{subsection.12.1.14}
\contentsline {subsection}{\numberline {12.1.15}\leavevmode {\color {Chapter }StringByInt}}{143}{subsection.12.1.15}
\contentsline {subsection}{\numberline {12.1.16}\leavevmode {\color {Chapter }PositionInTower}}{143}{subsection.12.1.16}
\contentsline {subsection}{\numberline {12.1.17}\leavevmode {\color {Chapter }RenameSubobjects}}{143}{subsection.12.1.17}
\contentsline {subsection}{\numberline {12.1.18}\leavevmode {\color {Chapter }CoefficientsInAbelianExtension}}{143}{subsection.12.1.18}
\contentsline {subsection}{\numberline {12.1.19}\leavevmode {\color {Chapter }MagmaEndomorphismByImagesNC}}{144}{subsection.12.1.19}
\contentsline {subsection}{\numberline {12.1.20}\leavevmode {\color {Chapter }MagmaHomomorphismByImagesNC}}{144}{subsection.12.1.20}
\contentsline {subsection}{\numberline {12.1.21}\leavevmode {\color {Chapter }NewFIFO}}{144}{subsection.12.1.21}
\contentsline {subsection}{\numberline {12.1.22}\leavevmode {\color {Chapter }ProductIdeal}}{144}{subsection.12.1.22}
\contentsline {subsection}{\numberline {12.1.23}\leavevmode {\color {Chapter }DimensionSeries}}{145}{subsection.12.1.23}
\contentsline {subsection}{\numberline {12.1.24}\leavevmode {\color {Chapter }Trans}}{145}{subsection.12.1.24}
\contentsline {section}{\numberline {12.2}\leavevmode {\color {Chapter }User settings}}{146}{section.12.2}
\contentsline {subsection}{\numberline {12.2.1}\leavevmode {\color {Chapter }InfoFR}}{146}{subsection.12.2.1}
\contentsline {subsection}{\numberline {12.2.2}\leavevmode {\color {Chapter }SEARCH@}}{146}{subsection.12.2.2}
