\contentsline {chapter}{\numberline {1}\textcolor {Chapter }{ The \textsf {Semigroups} package }}{8}{chapter.1}%
\contentsline {section}{\numberline {1.1}\textcolor {Chapter }{ Introduction }}{8}{section.1.1}%
\contentsline {section}{\numberline {1.2}\textcolor {Chapter }{ Overview }}{8}{section.1.2}%
\contentsline {chapter}{\numberline {2}\textcolor {Chapter }{Installing \textsf {Semigroups}}}{10}{chapter.2}%
\contentsline {section}{\numberline {2.1}\textcolor {Chapter }{For those in a hurry}}{10}{section.2.1}%
\contentsline {section}{\numberline {2.2}\textcolor {Chapter }{Compiling the kernel module}}{11}{section.2.2}%
\contentsline {section}{\numberline {2.3}\textcolor {Chapter }{Rebuilding the documentation}}{11}{section.2.3}%
\contentsline {section}{\numberline {2.4}\textcolor {Chapter }{Testing your installation}}{12}{section.2.4}%
\contentsline {subsection}{\numberline {2.4.1}\textcolor {Chapter }{SemigroupsTestInstall}}{12}{subsection.2.4.1}%
\contentsline {subsection}{\numberline {2.4.2}\textcolor {Chapter }{SemigroupsTestStandard}}{12}{subsection.2.4.2}%
\contentsline {subsection}{\numberline {2.4.3}\textcolor {Chapter }{SemigroupsTestExtreme}}{12}{subsection.2.4.3}%
\contentsline {subsection}{\numberline {2.4.4}\textcolor {Chapter }{SemigroupsTestAll}}{12}{subsection.2.4.4}%
\contentsline {section}{\numberline {2.5}\textcolor {Chapter }{More information during a computation}}{13}{section.2.5}%
\contentsline {subsection}{\numberline {2.5.1}\textcolor {Chapter }{InfoSemigroups}}{13}{subsection.2.5.1}%
\contentsline {chapter}{\numberline {3}\textcolor {Chapter }{ Bipartitions and blocks }}{14}{chapter.3}%
\contentsline {section}{\numberline {3.1}\textcolor {Chapter }{The family and categories of bipartitions}}{15}{section.3.1}%
\contentsline {subsection}{\numberline {3.1.1}\textcolor {Chapter }{IsBipartition}}{15}{subsection.3.1.1}%
\contentsline {subsection}{\numberline {3.1.2}\textcolor {Chapter }{IsBipartitionCollection}}{15}{subsection.3.1.2}%
\contentsline {section}{\numberline {3.2}\textcolor {Chapter }{Creating bipartitions}}{16}{section.3.2}%
\contentsline {subsection}{\numberline {3.2.1}\textcolor {Chapter }{Bipartition}}{16}{subsection.3.2.1}%
\contentsline {subsection}{\numberline {3.2.2}\textcolor {Chapter }{BipartitionByIntRep}}{16}{subsection.3.2.2}%
\contentsline {subsection}{\numberline {3.2.3}\textcolor {Chapter }{IdentityBipartition}}{16}{subsection.3.2.3}%
\contentsline {subsection}{\numberline {3.2.4}\textcolor {Chapter }{LeftOne (for a bipartition)}}{17}{subsection.3.2.4}%
\contentsline {subsection}{\numberline {3.2.5}\textcolor {Chapter }{RightOne (for a bipartition)}}{17}{subsection.3.2.5}%
\contentsline {subsection}{\numberline {3.2.6}\textcolor {Chapter }{StarOp (for a bipartition)}}{18}{subsection.3.2.6}%
\contentsline {subsection}{\numberline {3.2.7}\textcolor {Chapter }{RandomBipartition}}{18}{subsection.3.2.7}%
\contentsline {section}{\numberline {3.3}\textcolor {Chapter }{Changing the representation of a bipartition}}{18}{section.3.3}%
\contentsline {subsection}{\numberline {3.3.1}\textcolor {Chapter }{AsBipartition}}{19}{subsection.3.3.1}%
\contentsline {subsection}{\numberline {3.3.2}\textcolor {Chapter }{AsBlockBijection}}{20}{subsection.3.3.2}%
\contentsline {subsection}{\numberline {3.3.3}\textcolor {Chapter }{AsTransformation (for a bipartition)}}{21}{subsection.3.3.3}%
\contentsline {subsection}{\numberline {3.3.4}\textcolor {Chapter }{AsPartialPerm (for a bipartition)}}{21}{subsection.3.3.4}%
\contentsline {subsection}{\numberline {3.3.5}\textcolor {Chapter }{AsPermutation (for a bipartition)}}{22}{subsection.3.3.5}%
\contentsline {section}{\numberline {3.4}\textcolor {Chapter }{Operators for bipartitions}}{22}{section.3.4}%
\contentsline {subsection}{\numberline {3.4.1}\textcolor {Chapter }{PartialPermLeqBipartition}}{23}{subsection.3.4.1}%
\contentsline {subsection}{\numberline {3.4.2}\textcolor {Chapter }{NaturalLeqPartialPermBipartition}}{23}{subsection.3.4.2}%
\contentsline {subsection}{\numberline {3.4.3}\textcolor {Chapter }{NaturalLeqBlockBijection}}{23}{subsection.3.4.3}%
\contentsline {subsection}{\numberline {3.4.4}\textcolor {Chapter }{PermLeftQuoBipartition}}{23}{subsection.3.4.4}%
\contentsline {section}{\numberline {3.5}\textcolor {Chapter }{Attributes for bipartitons}}{24}{section.3.5}%
\contentsline {subsection}{\numberline {3.5.1}\textcolor {Chapter }{DegreeOfBipartition}}{24}{subsection.3.5.1}%
\contentsline {subsection}{\numberline {3.5.2}\textcolor {Chapter }{RankOfBipartition}}{24}{subsection.3.5.2}%
\contentsline {subsection}{\numberline {3.5.3}\textcolor {Chapter }{ExtRepOfObj (for a bipartition)}}{25}{subsection.3.5.3}%
\contentsline {subsection}{\numberline {3.5.4}\textcolor {Chapter }{IntRepOfBipartition}}{25}{subsection.3.5.4}%
\contentsline {subsection}{\numberline {3.5.5}\textcolor {Chapter }{RightBlocks}}{25}{subsection.3.5.5}%
\contentsline {subsection}{\numberline {3.5.6}\textcolor {Chapter }{LeftBlocks}}{26}{subsection.3.5.6}%
\contentsline {subsection}{\numberline {3.5.7}\textcolor {Chapter }{NrLeftBlocks}}{26}{subsection.3.5.7}%
\contentsline {subsection}{\numberline {3.5.8}\textcolor {Chapter }{NrRightBlocks}}{26}{subsection.3.5.8}%
\contentsline {subsection}{\numberline {3.5.9}\textcolor {Chapter }{NrBlocks (for blocks)}}{27}{subsection.3.5.9}%
\contentsline {subsection}{\numberline {3.5.10}\textcolor {Chapter }{DomainOfBipartition}}{27}{subsection.3.5.10}%
\contentsline {subsection}{\numberline {3.5.11}\textcolor {Chapter }{CodomainOfBipartition}}{27}{subsection.3.5.11}%
\contentsline {subsection}{\numberline {3.5.12}\textcolor {Chapter }{IsTransBipartition}}{28}{subsection.3.5.12}%
\contentsline {subsection}{\numberline {3.5.13}\textcolor {Chapter }{IsDualTransBipartition}}{28}{subsection.3.5.13}%
\contentsline {subsection}{\numberline {3.5.14}\textcolor {Chapter }{IsPermBipartition}}{29}{subsection.3.5.14}%
\contentsline {subsection}{\numberline {3.5.15}\textcolor {Chapter }{IsPartialPermBipartition}}{29}{subsection.3.5.15}%
\contentsline {subsection}{\numberline {3.5.16}\textcolor {Chapter }{IsBlockBijection}}{29}{subsection.3.5.16}%
\contentsline {subsection}{\numberline {3.5.17}\textcolor {Chapter }{IsUniformBlockBijection}}{30}{subsection.3.5.17}%
\contentsline {subsection}{\numberline {3.5.18}\textcolor {Chapter }{CanonicalBlocks}}{30}{subsection.3.5.18}%
\contentsline {section}{\numberline {3.6}\textcolor {Chapter }{Creating blocks and their attributes}}{30}{section.3.6}%
\contentsline {subsection}{\numberline {3.6.1}\textcolor {Chapter }{IsBlocks}}{31}{subsection.3.6.1}%
\contentsline {subsection}{\numberline {3.6.2}\textcolor {Chapter }{BLOCKS{\textunderscore }NC}}{31}{subsection.3.6.2}%
\contentsline {subsection}{\numberline {3.6.3}\textcolor {Chapter }{ExtRepOfObj (for a blocks)}}{31}{subsection.3.6.3}%
\contentsline {subsection}{\numberline {3.6.4}\textcolor {Chapter }{RankOfBlocks}}{31}{subsection.3.6.4}%
\contentsline {subsection}{\numberline {3.6.5}\textcolor {Chapter }{DegreeOfBlocks}}{32}{subsection.3.6.5}%
\contentsline {subsection}{\numberline {3.6.6}\textcolor {Chapter }{ProjectionFromBlocks}}{32}{subsection.3.6.6}%
\contentsline {section}{\numberline {3.7}\textcolor {Chapter }{Actions on blocks}}{32}{section.3.7}%
\contentsline {subsection}{\numberline {3.7.1}\textcolor {Chapter }{OnRightBlocks}}{32}{subsection.3.7.1}%
\contentsline {subsection}{\numberline {3.7.2}\textcolor {Chapter }{OnLeftBlocks}}{33}{subsection.3.7.2}%
\contentsline {section}{\numberline {3.8}\textcolor {Chapter }{ Semigroups of bipartitions }}{33}{section.3.8}%
\contentsline {subsection}{\numberline {3.8.1}\textcolor {Chapter }{IsBipartitionSemigroup}}{33}{subsection.3.8.1}%
\contentsline {subsection}{\numberline {3.8.2}\textcolor {Chapter }{IsBlockBijectionSemigroup}}{34}{subsection.3.8.2}%
\contentsline {subsection}{\numberline {3.8.3}\textcolor {Chapter }{IsPartialPermBipartitionSemigroup}}{34}{subsection.3.8.3}%
\contentsline {subsection}{\numberline {3.8.4}\textcolor {Chapter }{IsPermBipartitionGroup}}{34}{subsection.3.8.4}%
\contentsline {subsection}{\numberline {3.8.5}\textcolor {Chapter }{DegreeOfBipartitionSemigroup}}{35}{subsection.3.8.5}%
\contentsline {chapter}{\numberline {4}\textcolor {Chapter }{ Partitioned binary relations (PBRs) }}{36}{chapter.4}%
\contentsline {section}{\numberline {4.1}\textcolor {Chapter }{The family and categories of PBRs}}{36}{section.4.1}%
\contentsline {subsection}{\numberline {4.1.1}\textcolor {Chapter }{IsPBR}}{36}{subsection.4.1.1}%
\contentsline {subsection}{\numberline {4.1.2}\textcolor {Chapter }{IsPBRCollection}}{36}{subsection.4.1.2}%
\contentsline {section}{\numberline {4.2}\textcolor {Chapter }{Creating PBRs}}{36}{section.4.2}%
\contentsline {subsection}{\numberline {4.2.1}\textcolor {Chapter }{PBR}}{37}{subsection.4.2.1}%
\contentsline {subsection}{\numberline {4.2.2}\textcolor {Chapter }{RandomPBR}}{37}{subsection.4.2.2}%
\contentsline {subsection}{\numberline {4.2.3}\textcolor {Chapter }{EmptyPBR}}{37}{subsection.4.2.3}%
\contentsline {subsection}{\numberline {4.2.4}\textcolor {Chapter }{IdentityPBR}}{38}{subsection.4.2.4}%
\contentsline {subsection}{\numberline {4.2.5}\textcolor {Chapter }{UniversalPBR}}{38}{subsection.4.2.5}%
\contentsline {section}{\numberline {4.3}\textcolor {Chapter }{Changing the representation of a PBR}}{38}{section.4.3}%
\contentsline {subsection}{\numberline {4.3.1}\textcolor {Chapter }{AsPBR}}{38}{subsection.4.3.1}%
\contentsline {subsection}{\numberline {4.3.2}\textcolor {Chapter }{AsTransformation (for a PBR)}}{40}{subsection.4.3.2}%
\contentsline {subsection}{\numberline {4.3.3}\textcolor {Chapter }{AsPartialPerm (for a PBR)}}{40}{subsection.4.3.3}%
\contentsline {subsection}{\numberline {4.3.4}\textcolor {Chapter }{AsPermutation (for a PBR)}}{40}{subsection.4.3.4}%
\contentsline {section}{\numberline {4.4}\textcolor {Chapter }{Operators for PBRs}}{41}{section.4.4}%
\contentsline {section}{\numberline {4.5}\textcolor {Chapter }{Attributes for PBRs}}{41}{section.4.5}%
\contentsline {subsection}{\numberline {4.5.1}\textcolor {Chapter }{StarOp (for a PBR)}}{41}{subsection.4.5.1}%
\contentsline {subsection}{\numberline {4.5.2}\textcolor {Chapter }{DegreeOfPBR}}{41}{subsection.4.5.2}%
\contentsline {subsection}{\numberline {4.5.3}\textcolor {Chapter }{ExtRepOfObj (for a PBR)}}{42}{subsection.4.5.3}%
\contentsline {subsection}{\numberline {4.5.4}\textcolor {Chapter }{PBRNumber}}{42}{subsection.4.5.4}%
\contentsline {subsection}{\numberline {4.5.5}\textcolor {Chapter }{IsEmptyPBR}}{42}{subsection.4.5.5}%
\contentsline {subsection}{\numberline {4.5.6}\textcolor {Chapter }{IsIdentityPBR}}{43}{subsection.4.5.6}%
\contentsline {subsection}{\numberline {4.5.7}\textcolor {Chapter }{IsUniversalPBR}}{43}{subsection.4.5.7}%
\contentsline {subsection}{\numberline {4.5.8}\textcolor {Chapter }{IsBipartitionPBR}}{43}{subsection.4.5.8}%
\contentsline {subsection}{\numberline {4.5.9}\textcolor {Chapter }{IsTransformationPBR}}{44}{subsection.4.5.9}%
\contentsline {subsection}{\numberline {4.5.10}\textcolor {Chapter }{IsDualTransformationPBR}}{44}{subsection.4.5.10}%
\contentsline {subsection}{\numberline {4.5.11}\textcolor {Chapter }{IsPartialPermPBR}}{45}{subsection.4.5.11}%
\contentsline {subsection}{\numberline {4.5.12}\textcolor {Chapter }{IsPermPBR}}{45}{subsection.4.5.12}%
\contentsline {section}{\numberline {4.6}\textcolor {Chapter }{ Semigroups of PBRs }}{46}{section.4.6}%
\contentsline {subsection}{\numberline {4.6.1}\textcolor {Chapter }{IsPBRSemigroup}}{46}{subsection.4.6.1}%
\contentsline {subsection}{\numberline {4.6.2}\textcolor {Chapter }{DegreeOfPBRSemigroup}}{47}{subsection.4.6.2}%
\contentsline {chapter}{\numberline {5}\textcolor {Chapter }{ Matrices over semirings }}{48}{chapter.5}%
\contentsline {section}{\numberline {5.1}\textcolor {Chapter }{Creating matrices over semirings}}{49}{section.5.1}%
\contentsline {subsection}{\numberline {5.1.1}\textcolor {Chapter }{IsMatrixOverSemiring}}{49}{subsection.5.1.1}%
\contentsline {subsection}{\numberline {5.1.2}\textcolor {Chapter }{IsMatrixOverSemiringCollection}}{49}{subsection.5.1.2}%
\contentsline {subsection}{\numberline {5.1.3}\textcolor {Chapter }{DimensionOfMatrixOverSemiring}}{49}{subsection.5.1.3}%
\contentsline {subsection}{\numberline {5.1.4}\textcolor {Chapter }{DimensionOfMatrixOverSemiringCollection}}{50}{subsection.5.1.4}%
\contentsline {subsection}{\numberline {5.1.5}\textcolor {Chapter }{Matrix (for a filter and a matrix)}}{50}{subsection.5.1.5}%
\contentsline {subsection}{\numberline {5.1.6}\textcolor {Chapter }{AsMatrix (for a filter and a matrix)}}{51}{subsection.5.1.6}%
\contentsline {subsection}{\numberline {5.1.7}\textcolor {Chapter }{RandomMatrix (for a filter and a matrix)}}{53}{subsection.5.1.7}%
\contentsline {subsection}{\numberline {5.1.8}\textcolor {Chapter }{Matrix filters}}{54}{subsection.5.1.8}%
\contentsline {subsection}{\numberline {5.1.9}\textcolor {Chapter }{Matrix collection filters}}{54}{subsection.5.1.9}%
\contentsline {subsection}{\numberline {5.1.10}\textcolor {Chapter }{AsList}}{55}{subsection.5.1.10}%
\contentsline {subsection}{\numberline {5.1.11}\textcolor {Chapter }{ThresholdTropicalMatrix}}{55}{subsection.5.1.11}%
\contentsline {subsection}{\numberline {5.1.12}\textcolor {Chapter }{ThresholdNTPMatrix}}{56}{subsection.5.1.12}%
\contentsline {section}{\numberline {5.2}\textcolor {Chapter }{Operators for matrices over semirings}}{57}{section.5.2}%
\contentsline {section}{\numberline {5.3}\textcolor {Chapter }{ Boolean matrices }}{57}{section.5.3}%
\contentsline {subsection}{\numberline {5.3.1}\textcolor {Chapter }{BooleanMat}}{57}{subsection.5.3.1}%
\contentsline {subsection}{\numberline {5.3.2}\textcolor {Chapter }{AsBooleanMat}}{58}{subsection.5.3.2}%
\contentsline {subsection}{\numberline {5.3.3}\textcolor {Chapter }{\texttt {\symbol {92}}in}}{61}{subsection.5.3.3}%
\contentsline {subsection}{\numberline {5.3.4}\textcolor {Chapter }{OnBlist}}{61}{subsection.5.3.4}%
\contentsline {subsection}{\numberline {5.3.5}\textcolor {Chapter }{Successors}}{61}{subsection.5.3.5}%
\contentsline {subsection}{\numberline {5.3.6}\textcolor {Chapter }{BooleanMatNumber}}{62}{subsection.5.3.6}%
\contentsline {subsection}{\numberline {5.3.7}\textcolor {Chapter }{BlistNumber}}{62}{subsection.5.3.7}%
\contentsline {subsection}{\numberline {5.3.8}\textcolor {Chapter }{CanonicalBooleanMat (for a perm group, perm group and boolean matrix)}}{63}{subsection.5.3.8}%
\contentsline {subsection}{\numberline {5.3.9}\textcolor {Chapter }{IsRowTrimBooleanMat}}{64}{subsection.5.3.9}%
\contentsline {subsection}{\numberline {5.3.10}\textcolor {Chapter }{IsSymmetricBooleanMat}}{64}{subsection.5.3.10}%
\contentsline {subsection}{\numberline {5.3.11}\textcolor {Chapter }{IsReflexiveBooleanMat}}{65}{subsection.5.3.11}%
\contentsline {subsection}{\numberline {5.3.12}\textcolor {Chapter }{IsTransitiveBooleanMat}}{65}{subsection.5.3.12}%
\contentsline {subsection}{\numberline {5.3.13}\textcolor {Chapter }{IsAntiSymmetricBooleanMat}}{66}{subsection.5.3.13}%
\contentsline {subsection}{\numberline {5.3.14}\textcolor {Chapter }{IsTotalBooleanMat}}{66}{subsection.5.3.14}%
\contentsline {subsection}{\numberline {5.3.15}\textcolor {Chapter }{IsPartialOrderBooleanMat}}{67}{subsection.5.3.15}%
\contentsline {subsection}{\numberline {5.3.16}\textcolor {Chapter }{IsEquivalenceBooleanMat}}{67}{subsection.5.3.16}%
\contentsline {subsection}{\numberline {5.3.17}\textcolor {Chapter }{IsTransformationBooleanMat}}{67}{subsection.5.3.17}%
\contentsline {section}{\numberline {5.4}\textcolor {Chapter }{ Matrices over finite fields }}{67}{section.5.4}%
\contentsline {subsection}{\numberline {5.4.1}\textcolor {Chapter }{RowSpaceBasis (for a matrix over finite field)}}{68}{subsection.5.4.1}%
\contentsline {subsection}{\numberline {5.4.2}\textcolor {Chapter }{RightInverse (for a matrix over finite field)}}{68}{subsection.5.4.2}%
\contentsline {section}{\numberline {5.5}\textcolor {Chapter }{ Matrices over the integers }}{68}{section.5.5}%
\contentsline {subsection}{\numberline {5.5.1}\textcolor {Chapter }{InverseOp (for an integer matrix)}}{69}{subsection.5.5.1}%
\contentsline {subsection}{\numberline {5.5.2}\textcolor {Chapter }{IsTorsion (for an integer matrix)}}{69}{subsection.5.5.2}%
\contentsline {subsection}{\numberline {5.5.3}\textcolor {Chapter }{Order}}{69}{subsection.5.5.3}%
\contentsline {section}{\numberline {5.6}\textcolor {Chapter }{ Max\texttt {\symbol {45}}plus and min\texttt {\symbol {45}}plus matrices }}{70}{section.5.6}%
\contentsline {subsection}{\numberline {5.6.1}\textcolor {Chapter }{InverseOp}}{70}{subsection.5.6.1}%
\contentsline {subsection}{\numberline {5.6.2}\textcolor {Chapter }{RadialEigenvector}}{70}{subsection.5.6.2}%
\contentsline {subsection}{\numberline {5.6.3}\textcolor {Chapter }{SpectralRadius}}{71}{subsection.5.6.3}%
\contentsline {subsection}{\numberline {5.6.4}\textcolor {Chapter }{UnweightedPrecedenceDigraph}}{71}{subsection.5.6.4}%
\contentsline {section}{\numberline {5.7}\textcolor {Chapter }{ Matrix semigroups }}{71}{section.5.7}%
\contentsline {subsection}{\numberline {5.7.1}\textcolor {Chapter }{Matrix semigroup filters}}{72}{subsection.5.7.1}%
\contentsline {subsection}{\numberline {5.7.2}\textcolor {Chapter }{Matrix monoid filters}}{72}{subsection.5.7.2}%
\contentsline {subsection}{\numberline {5.7.3}\textcolor {Chapter }{IsFinite}}{72}{subsection.5.7.3}%
\contentsline {subsection}{\numberline {5.7.4}\textcolor {Chapter }{IsTorsion}}{73}{subsection.5.7.4}%
\contentsline {subsection}{\numberline {5.7.5}\textcolor {Chapter }{NormalizeSemigroup}}{73}{subsection.5.7.5}%
\contentsline {chapter}{\numberline {6}\textcolor {Chapter }{ Semigroups and monoids defined by generating sets }}{74}{chapter.6}%
\contentsline {section}{\numberline {6.1}\textcolor {Chapter }{Underlying algorithms}}{74}{section.6.1}%
\contentsline {subsection}{\numberline {6.1.1}\textcolor {Chapter }{ Acting semigroups }}{74}{subsection.6.1.1}%
\contentsline {subsection}{\numberline {6.1.2}\textcolor {Chapter }{IsActingSemigroup}}{75}{subsection.6.1.2}%
\contentsline {subsection}{\numberline {6.1.3}\textcolor {Chapter }{The Froidure\texttt {\symbol {45}}Pin Algorithm}}{75}{subsection.6.1.3}%
\contentsline {subsection}{\numberline {6.1.4}\textcolor {Chapter }{CanUseFroidurePin}}{76}{subsection.6.1.4}%
\contentsline {section}{\numberline {6.2}\textcolor {Chapter }{Semigroups represented by generators}}{77}{section.6.2}%
\contentsline {subsection}{\numberline {6.2.1}\textcolor {Chapter }{InverseMonoidByGenerators}}{77}{subsection.6.2.1}%
\contentsline {section}{\numberline {6.3}\textcolor {Chapter }{Options when creating semigroups}}{77}{section.6.3}%
\contentsline {subsection}{\numberline {6.3.1}\textcolor {Chapter }{SEMIGROUPS.DefaultOptionsRec}}{79}{subsection.6.3.1}%
\contentsline {section}{\numberline {6.4}\textcolor {Chapter }{Subsemigroups and supersemigroups}}{79}{section.6.4}%
\contentsline {subsection}{\numberline {6.4.1}\textcolor {Chapter }{ClosureSemigroup}}{79}{subsection.6.4.1}%
\contentsline {subsection}{\numberline {6.4.2}\textcolor {Chapter }{SubsemigroupByProperty (for a semigroup and function)}}{81}{subsection.6.4.2}%
\contentsline {subsection}{\numberline {6.4.3}\textcolor {Chapter }{InverseSubsemigroupByProperty}}{81}{subsection.6.4.3}%
\contentsline {section}{\numberline {6.5}\textcolor {Chapter }{Changing the representation of a semigroup}}{82}{section.6.5}%
\contentsline {subsection}{\numberline {6.5.1}\textcolor {Chapter }{IsomorphismSemigroup}}{83}{subsection.6.5.1}%
\contentsline {subsection}{\numberline {6.5.2}\textcolor {Chapter }{IsomorphismMonoid}}{84}{subsection.6.5.2}%
\contentsline {subsection}{\numberline {6.5.3}\textcolor {Chapter }{AsSemigroup}}{85}{subsection.6.5.3}%
\contentsline {subsection}{\numberline {6.5.4}\textcolor {Chapter }{AsMonoid}}{86}{subsection.6.5.4}%
\contentsline {subsection}{\numberline {6.5.5}\textcolor {Chapter }{IsomorphismPermGroup}}{87}{subsection.6.5.5}%
\contentsline {subsection}{\numberline {6.5.6}\textcolor {Chapter }{RZMSNormalization}}{87}{subsection.6.5.6}%
\contentsline {subsection}{\numberline {6.5.7}\textcolor {Chapter }{RMSNormalization}}{89}{subsection.6.5.7}%
\contentsline {subsection}{\numberline {6.5.8}\textcolor {Chapter }{IsomorphismReesMatrixSemigroup (for a semigroup)}}{89}{subsection.6.5.8}%
\contentsline {subsection}{\numberline {6.5.9}\textcolor {Chapter }{AntiIsomorphismDualFpSemigroup}}{90}{subsection.6.5.9}%
\contentsline {subsection}{\numberline {6.5.10}\textcolor {Chapter }{EmbeddingFpMonoid}}{90}{subsection.6.5.10}%
\contentsline {section}{\numberline {6.6}\textcolor {Chapter }{Random semigroups}}{91}{section.6.6}%
\contentsline {subsection}{\numberline {6.6.1}\textcolor {Chapter }{RandomSemigroup}}{91}{subsection.6.6.1}%
\contentsline {chapter}{\numberline {7}\textcolor {Chapter }{ Standard examples }}{94}{chapter.7}%
\contentsline {section}{\numberline {7.1}\textcolor {Chapter }{ Transformation semigroups }}{94}{section.7.1}%
\contentsline {subsection}{\numberline {7.1.1}\textcolor {Chapter }{CatalanMonoid}}{94}{subsection.7.1.1}%
\contentsline {subsection}{\numberline {7.1.2}\textcolor {Chapter }{EndomorphismsPartition}}{94}{subsection.7.1.2}%
\contentsline {subsection}{\numberline {7.1.3}\textcolor {Chapter }{PartialTransformationMonoid}}{95}{subsection.7.1.3}%
\contentsline {subsection}{\numberline {7.1.4}\textcolor {Chapter }{SingularTransformationSemigroup}}{95}{subsection.7.1.4}%
\contentsline {subsection}{\numberline {7.1.5}\textcolor {Chapter }{Semigroups of order\texttt {\symbol {45}}preserving transformations}}{95}{subsection.7.1.5}%
\contentsline {subsection}{\numberline {7.1.6}\textcolor {Chapter }{EndomorphismMonoid (for a digraph)}}{97}{subsection.7.1.6}%
\contentsline {section}{\numberline {7.2}\textcolor {Chapter }{ Semigroups of partial permutations }}{97}{section.7.2}%
\contentsline {subsection}{\numberline {7.2.1}\textcolor {Chapter }{MunnSemigroup}}{98}{subsection.7.2.1}%
\contentsline {subsection}{\numberline {7.2.2}\textcolor {Chapter }{RookMonoid}}{98}{subsection.7.2.2}%
\contentsline {subsection}{\numberline {7.2.3}\textcolor {Chapter }{Inverse monoids of order\texttt {\symbol {45}}preserving partial permutations}}{98}{subsection.7.2.3}%
\contentsline {section}{\numberline {7.3}\textcolor {Chapter }{ Semigroups of bipartitions }}{99}{section.7.3}%
\contentsline {subsection}{\numberline {7.3.1}\textcolor {Chapter }{PartitionMonoid}}{99}{subsection.7.3.1}%
\contentsline {subsection}{\numberline {7.3.2}\textcolor {Chapter }{BrauerMonoid}}{100}{subsection.7.3.2}%
\contentsline {subsection}{\numberline {7.3.3}\textcolor {Chapter }{JonesMonoid}}{101}{subsection.7.3.3}%
\contentsline {subsection}{\numberline {7.3.4}\textcolor {Chapter }{PartialJonesMonoid}}{101}{subsection.7.3.4}%
\contentsline {subsection}{\numberline {7.3.5}\textcolor {Chapter }{AnnularJonesMonoid}}{102}{subsection.7.3.5}%
\contentsline {subsection}{\numberline {7.3.6}\textcolor {Chapter }{MotzkinMonoid}}{102}{subsection.7.3.6}%
\contentsline {subsection}{\numberline {7.3.7}\textcolor {Chapter }{DualSymmetricInverseSemigroup}}{102}{subsection.7.3.7}%
\contentsline {subsection}{\numberline {7.3.8}\textcolor {Chapter }{UniformBlockBijectionMonoid}}{103}{subsection.7.3.8}%
\contentsline {subsection}{\numberline {7.3.9}\textcolor {Chapter }{PlanarPartitionMonoid}}{104}{subsection.7.3.9}%
\contentsline {subsection}{\numberline {7.3.10}\textcolor {Chapter }{ModularPartitionMonoid}}{105}{subsection.7.3.10}%
\contentsline {subsection}{\numberline {7.3.11}\textcolor {Chapter }{ApsisMonoid}}{105}{subsection.7.3.11}%
\contentsline {section}{\numberline {7.4}\textcolor {Chapter }{ Standard PBR semigroups }}{106}{section.7.4}%
\contentsline {subsection}{\numberline {7.4.1}\textcolor {Chapter }{FullPBRMonoid}}{106}{subsection.7.4.1}%
\contentsline {section}{\numberline {7.5}\textcolor {Chapter }{ Semigroups of matrices over a finite field }}{106}{section.7.5}%
\contentsline {subsection}{\numberline {7.5.1}\textcolor {Chapter }{FullMatrixMonoid}}{107}{subsection.7.5.1}%
\contentsline {subsection}{\numberline {7.5.2}\textcolor {Chapter }{SpecialLinearMonoid}}{107}{subsection.7.5.2}%
\contentsline {subsection}{\numberline {7.5.3}\textcolor {Chapter }{IsFullMatrixMonoid}}{107}{subsection.7.5.3}%
\contentsline {section}{\numberline {7.6}\textcolor {Chapter }{ Semigroups of boolean matrices }}{108}{section.7.6}%
\contentsline {subsection}{\numberline {7.6.1}\textcolor {Chapter }{FullBooleanMatMonoid}}{108}{subsection.7.6.1}%
\contentsline {subsection}{\numberline {7.6.2}\textcolor {Chapter }{RegularBooleanMatMonoid}}{108}{subsection.7.6.2}%
\contentsline {subsection}{\numberline {7.6.3}\textcolor {Chapter }{ReflexiveBooleanMatMonoid}}{109}{subsection.7.6.3}%
\contentsline {subsection}{\numberline {7.6.4}\textcolor {Chapter }{HallMonoid}}{109}{subsection.7.6.4}%
\contentsline {subsection}{\numberline {7.6.5}\textcolor {Chapter }{GossipMonoid}}{109}{subsection.7.6.5}%
\contentsline {subsection}{\numberline {7.6.6}\textcolor {Chapter }{TriangularBooleanMatMonoid}}{110}{subsection.7.6.6}%
\contentsline {section}{\numberline {7.7}\textcolor {Chapter }{ Semigroups of matrices over a semiring }}{110}{section.7.7}%
\contentsline {subsection}{\numberline {7.7.1}\textcolor {Chapter }{FullTropicalMaxPlusMonoid}}{110}{subsection.7.7.1}%
\contentsline {subsection}{\numberline {7.7.2}\textcolor {Chapter }{FullTropicalMinPlusMonoid}}{111}{subsection.7.7.2}%
\contentsline {section}{\numberline {7.8}\textcolor {Chapter }{ Examples in various representations }}{111}{section.7.8}%
\contentsline {subsection}{\numberline {7.8.1}\textcolor {Chapter }{TrivialSemigroup}}{111}{subsection.7.8.1}%
\contentsline {subsection}{\numberline {7.8.2}\textcolor {Chapter }{MonogenicSemigroup}}{112}{subsection.7.8.2}%
\contentsline {subsection}{\numberline {7.8.3}\textcolor {Chapter }{RectangularBand}}{112}{subsection.7.8.3}%
\contentsline {subsection}{\numberline {7.8.4}\textcolor {Chapter }{FreeSemilattice}}{113}{subsection.7.8.4}%
\contentsline {subsection}{\numberline {7.8.5}\textcolor {Chapter }{ZeroSemigroup}}{114}{subsection.7.8.5}%
\contentsline {subsection}{\numberline {7.8.6}\textcolor {Chapter }{LeftZeroSemigroup}}{115}{subsection.7.8.6}%
\contentsline {subsection}{\numberline {7.8.7}\textcolor {Chapter }{BrandtSemigroup}}{116}{subsection.7.8.7}%
\contentsline {section}{\numberline {7.9}\textcolor {Chapter }{ Free bands }}{117}{section.7.9}%
\contentsline {subsection}{\numberline {7.9.1}\textcolor {Chapter }{FreeBand (for a given rank)}}{117}{subsection.7.9.1}%
\contentsline {subsection}{\numberline {7.9.2}\textcolor {Chapter }{IsFreeBandCategory}}{118}{subsection.7.9.2}%
\contentsline {subsection}{\numberline {7.9.3}\textcolor {Chapter }{IsFreeBand (for a given semigroup)}}{118}{subsection.7.9.3}%
\contentsline {subsection}{\numberline {7.9.4}\textcolor {Chapter }{IsFreeBandElement}}{118}{subsection.7.9.4}%
\contentsline {subsection}{\numberline {7.9.5}\textcolor {Chapter }{IsFreeBandElementCollection}}{118}{subsection.7.9.5}%
\contentsline {subsection}{\numberline {7.9.6}\textcolor {Chapter }{IsFreeBandSubsemigroup}}{118}{subsection.7.9.6}%
\contentsline {subsection}{\numberline {7.9.7}\textcolor {Chapter }{ContentOfFreeBandElement}}{119}{subsection.7.9.7}%
\contentsline {subsection}{\numberline {7.9.8}\textcolor {Chapter }{EqualInFreeBand}}{119}{subsection.7.9.8}%
\contentsline {subsection}{\numberline {7.9.9}\textcolor {Chapter }{GreensDClassOfElement (for a free band and element)}}{120}{subsection.7.9.9}%
\contentsline {subsection}{\numberline {7.9.10}\textcolor {Chapter }{Operators}}{120}{subsection.7.9.10}%
\contentsline {section}{\numberline {7.10}\textcolor {Chapter }{ Graph inverse semigroups }}{120}{section.7.10}%
\contentsline {subsection}{\numberline {7.10.1}\textcolor {Chapter }{GraphInverseSemigroup}}{121}{subsection.7.10.1}%
\contentsline {subsection}{\numberline {7.10.2}\textcolor {Chapter }{Range (for a graph inverse semigroup element)}}{122}{subsection.7.10.2}%
\contentsline {subsection}{\numberline {7.10.3}\textcolor {Chapter }{IsVertex (for a graph inverse semigroup element)}}{122}{subsection.7.10.3}%
\contentsline {subsection}{\numberline {7.10.4}\textcolor {Chapter }{IsGraphInverseSemigroup}}{122}{subsection.7.10.4}%
\contentsline {subsection}{\numberline {7.10.5}\textcolor {Chapter }{GraphOfGraphInverseSemigroup}}{123}{subsection.7.10.5}%
\contentsline {subsection}{\numberline {7.10.6}\textcolor {Chapter }{IsGraphInverseSemigroupElementCollection}}{123}{subsection.7.10.6}%
\contentsline {subsection}{\numberline {7.10.7}\textcolor {Chapter }{IsGraphInverseSubsemigroup}}{123}{subsection.7.10.7}%
\contentsline {subsection}{\numberline {7.10.8}\textcolor {Chapter }{VerticesOfGraphInverseSemigroup}}{124}{subsection.7.10.8}%
\contentsline {subsection}{\numberline {7.10.9}\textcolor {Chapter }{IndexOfVertexOfGraphInverseSemigroup}}{124}{subsection.7.10.9}%
\contentsline {section}{\numberline {7.11}\textcolor {Chapter }{ Free inverse semigroups }}{124}{section.7.11}%
\contentsline {subsection}{\numberline {7.11.1}\textcolor {Chapter }{FreeInverseSemigroup (for a given rank)}}{125}{subsection.7.11.1}%
\contentsline {subsection}{\numberline {7.11.2}\textcolor {Chapter }{IsFreeInverseSemigroupCategory}}{126}{subsection.7.11.2}%
\contentsline {subsection}{\numberline {7.11.3}\textcolor {Chapter }{IsFreeInverseSemigroup}}{126}{subsection.7.11.3}%
\contentsline {subsection}{\numberline {7.11.4}\textcolor {Chapter }{IsFreeInverseSemigroupElement}}{126}{subsection.7.11.4}%
\contentsline {subsection}{\numberline {7.11.5}\textcolor {Chapter }{IsFreeInverseSemigroupElementCollection}}{126}{subsection.7.11.5}%
\contentsline {subsection}{\numberline {7.11.6}\textcolor {Chapter }{CanonicalForm (for a free inverse semigroup element)}}{126}{subsection.7.11.6}%
\contentsline {subsection}{\numberline {7.11.7}\textcolor {Chapter }{MinimalWord (for free inverse semigroup element)}}{127}{subsection.7.11.7}%
\contentsline {subsection}{\numberline {7.11.8}\textcolor {Chapter }{Displaying free inverse semigroup elements }}{127}{subsection.7.11.8}%
\contentsline {subsection}{\numberline {7.11.9}\textcolor {Chapter }{Operators for free inverse semigroup elements }}{128}{subsection.7.11.9}%
\contentsline {chapter}{\numberline {8}\textcolor {Chapter }{ Standard constructions }}{129}{chapter.8}%
\contentsline {section}{\numberline {8.1}\textcolor {Chapter }{ Products of semigroups }}{129}{section.8.1}%
\contentsline {subsection}{\numberline {8.1.1}\textcolor {Chapter }{DirectProduct}}{129}{subsection.8.1.1}%
\contentsline {subsection}{\numberline {8.1.2}\textcolor {Chapter }{WreathProduct}}{130}{subsection.8.1.2}%
\contentsline {section}{\numberline {8.2}\textcolor {Chapter }{ Dual semigroups }}{130}{section.8.2}%
\contentsline {subsection}{\numberline {8.2.1}\textcolor {Chapter }{DualSemigroup}}{130}{subsection.8.2.1}%
\contentsline {subsection}{\numberline {8.2.2}\textcolor {Chapter }{IsDualSemigroupRep}}{131}{subsection.8.2.2}%
\contentsline {subsection}{\numberline {8.2.3}\textcolor {Chapter }{IsDualSemigroupElement}}{131}{subsection.8.2.3}%
\contentsline {subsection}{\numberline {8.2.4}\textcolor {Chapter }{AntiIsomorphismDualSemigroup}}{132}{subsection.8.2.4}%
\contentsline {section}{\numberline {8.3}\textcolor {Chapter }{ Strong semilattices of semigroups }}{132}{section.8.3}%
\contentsline {subsection}{\numberline {8.3.1}\textcolor {Chapter }{StrongSemilatticeOfSemigroups}}{133}{subsection.8.3.1}%
\contentsline {subsection}{\numberline {8.3.2}\textcolor {Chapter }{SSSE}}{133}{subsection.8.3.2}%
\contentsline {subsection}{\numberline {8.3.3}\textcolor {Chapter }{IsSSSE}}{134}{subsection.8.3.3}%
\contentsline {subsection}{\numberline {8.3.4}\textcolor {Chapter }{IsStrongSemilatticeOfSemigroups}}{134}{subsection.8.3.4}%
\contentsline {subsection}{\numberline {8.3.5}\textcolor {Chapter }{SemilatticeOfStrongSemilatticeOfSemigroups}}{134}{subsection.8.3.5}%
\contentsline {subsection}{\numberline {8.3.6}\textcolor {Chapter }{SemigroupsOfStrongSemilatticeOfSemigroups}}{134}{subsection.8.3.6}%
\contentsline {subsection}{\numberline {8.3.7}\textcolor {Chapter }{HomomorphismsOfStrongSemilatticeOfSemigroups}}{135}{subsection.8.3.7}%
\contentsline {section}{\numberline {8.4}\textcolor {Chapter }{ McAlister triple semigroups }}{135}{section.8.4}%
\contentsline {subsection}{\numberline {8.4.1}\textcolor {Chapter }{IsMcAlisterTripleSemigroup}}{136}{subsection.8.4.1}%
\contentsline {subsection}{\numberline {8.4.2}\textcolor {Chapter }{McAlisterTripleSemigroup}}{136}{subsection.8.4.2}%
\contentsline {subsection}{\numberline {8.4.3}\textcolor {Chapter }{McAlisterTripleSemigroupGroup}}{137}{subsection.8.4.3}%
\contentsline {subsection}{\numberline {8.4.4}\textcolor {Chapter }{McAlisterTripleSemigroupPartialOrder}}{137}{subsection.8.4.4}%
\contentsline {subsection}{\numberline {8.4.5}\textcolor {Chapter }{McAlisterTripleSemigroupSemilattice}}{137}{subsection.8.4.5}%
\contentsline {subsection}{\numberline {8.4.6}\textcolor {Chapter }{McAlisterTripleSemigroupAction}}{138}{subsection.8.4.6}%
\contentsline {subsection}{\numberline {8.4.7}\textcolor {Chapter }{IsMcAlisterTripleSemigroupElement}}{138}{subsection.8.4.7}%
\contentsline {subsection}{\numberline {8.4.8}\textcolor {Chapter }{McAlisterTripleSemigroupElement}}{138}{subsection.8.4.8}%
\contentsline {chapter}{\numberline {9}\textcolor {Chapter }{ Ideals }}{140}{chapter.9}%
\contentsline {section}{\numberline {9.1}\textcolor {Chapter }{ Creating ideals }}{140}{section.9.1}%
\contentsline {subsection}{\numberline {9.1.1}\textcolor {Chapter }{SemigroupIdeal}}{140}{subsection.9.1.1}%
\contentsline {subsection}{\numberline {9.1.2}\textcolor {Chapter }{Ideals (for a semigroup)}}{141}{subsection.9.1.2}%
\contentsline {section}{\numberline {9.2}\textcolor {Chapter }{ Attributes of ideals }}{141}{section.9.2}%
\contentsline {subsection}{\numberline {9.2.1}\textcolor {Chapter }{GeneratorsOfSemigroupIdeal}}{141}{subsection.9.2.1}%
\contentsline {subsection}{\numberline {9.2.2}\textcolor {Chapter }{MinimalIdealGeneratingSet}}{142}{subsection.9.2.2}%
\contentsline {subsection}{\numberline {9.2.3}\textcolor {Chapter }{SupersemigroupOfIdeal}}{142}{subsection.9.2.3}%
\contentsline {chapter}{\numberline {10}\textcolor {Chapter }{ Green's relations }}{144}{chapter.10}%
\contentsline {section}{\numberline {10.1}\textcolor {Chapter }{ Creating Green's classes and representatives }}{144}{section.10.1}%
\contentsline {subsection}{\numberline {10.1.1}\textcolor {Chapter }{XClassOfYClass}}{144}{subsection.10.1.1}%
\contentsline {subsection}{\numberline {10.1.2}\textcolor {Chapter }{GreensXClassOfElement}}{145}{subsection.10.1.2}%
\contentsline {subsection}{\numberline {10.1.3}\textcolor {Chapter }{GreensXClassOfElementNC}}{146}{subsection.10.1.3}%
\contentsline {subsection}{\numberline {10.1.4}\textcolor {Chapter }{GreensXClasses}}{147}{subsection.10.1.4}%
\contentsline {subsection}{\numberline {10.1.5}\textcolor {Chapter }{XClassReps}}{149}{subsection.10.1.5}%
\contentsline {subsection}{\numberline {10.1.6}\textcolor {Chapter }{MinimalDClass}}{150}{subsection.10.1.6}%
\contentsline {subsection}{\numberline {10.1.7}\textcolor {Chapter }{MaximalXClasses}}{150}{subsection.10.1.7}%
\contentsline {subsection}{\numberline {10.1.8}\textcolor {Chapter }{NrRegularDClasses}}{151}{subsection.10.1.8}%
\contentsline {subsection}{\numberline {10.1.9}\textcolor {Chapter }{NrXClasses}}{151}{subsection.10.1.9}%
\contentsline {subsection}{\numberline {10.1.10}\textcolor {Chapter }{PartialOrderOfXClasses}}{152}{subsection.10.1.10}%
\contentsline {subsection}{\numberline {10.1.11}\textcolor {Chapter }{LengthOfLongestDClassChain}}{153}{subsection.10.1.11}%
\contentsline {subsection}{\numberline {10.1.12}\textcolor {Chapter }{IsGreensDGreaterThanFunc}}{154}{subsection.10.1.12}%
\contentsline {section}{\numberline {10.2}\textcolor {Chapter }{ Iterators and enumerators of classes and representatives }}{155}{section.10.2}%
\contentsline {subsection}{\numberline {10.2.1}\textcolor {Chapter }{IteratorOfXClassReps}}{155}{subsection.10.2.1}%
\contentsline {subsection}{\numberline {10.2.2}\textcolor {Chapter }{IteratorOfXClasses}}{155}{subsection.10.2.2}%
\contentsline {section}{\numberline {10.3}\textcolor {Chapter }{ Properties of Green's classes }}{156}{section.10.3}%
\contentsline {subsection}{\numberline {10.3.1}\textcolor {Chapter }{Less than for Green's classes}}{157}{subsection.10.3.1}%
\contentsline {subsection}{\numberline {10.3.2}\textcolor {Chapter }{IsRegularGreensClass}}{157}{subsection.10.3.2}%
\contentsline {subsection}{\numberline {10.3.3}\textcolor {Chapter }{IsGreensClassNC}}{158}{subsection.10.3.3}%
\contentsline {section}{\numberline {10.4}\textcolor {Chapter }{ Attributes of Green's classes }}{158}{section.10.4}%
\contentsline {subsection}{\numberline {10.4.1}\textcolor {Chapter }{GroupHClass}}{158}{subsection.10.4.1}%
\contentsline {subsection}{\numberline {10.4.2}\textcolor {Chapter }{SchutzenbergerGroup}}{159}{subsection.10.4.2}%
\contentsline {subsection}{\numberline {10.4.3}\textcolor {Chapter }{StructureDescriptionSchutzenbergerGroups}}{160}{subsection.10.4.3}%
\contentsline {subsection}{\numberline {10.4.4}\textcolor {Chapter }{StructureDescriptionMaximalSubgroups}}{161}{subsection.10.4.4}%
\contentsline {subsection}{\numberline {10.4.5}\textcolor {Chapter }{MultiplicativeNeutralElement (for an H-class)}}{161}{subsection.10.4.5}%
\contentsline {subsection}{\numberline {10.4.6}\textcolor {Chapter }{StructureDescription (for an H-class)}}{162}{subsection.10.4.6}%
\contentsline {subsection}{\numberline {10.4.7}\textcolor {Chapter }{InjectionPrincipalFactor}}{162}{subsection.10.4.7}%
\contentsline {subsection}{\numberline {10.4.8}\textcolor {Chapter }{PrincipalFactor}}{163}{subsection.10.4.8}%
\contentsline {section}{\numberline {10.5}\textcolor {Chapter }{ Operations for Green's relations and classes }}{164}{section.10.5}%
\contentsline {subsection}{\numberline {10.5.1}\textcolor {Chapter }{LeftGreensMultiplier}}{164}{subsection.10.5.1}%
\contentsline {chapter}{\numberline {11}\textcolor {Chapter }{ Attributes and operations for semigroups }}{166}{chapter.11}%
\contentsline {section}{\numberline {11.1}\textcolor {Chapter }{ Accessing the elements of a semigroup }}{166}{section.11.1}%
\contentsline {subsection}{\numberline {11.1.1}\textcolor {Chapter }{AsListCanonical}}{166}{subsection.11.1.1}%
\contentsline {subsection}{\numberline {11.1.2}\textcolor {Chapter }{PositionCanonical}}{167}{subsection.11.1.2}%
\contentsline {subsection}{\numberline {11.1.3}\textcolor {Chapter }{Enumerate}}{167}{subsection.11.1.3}%
\contentsline {subsection}{\numberline {11.1.4}\textcolor {Chapter }{IsEnumerated}}{168}{subsection.11.1.4}%
\contentsline {section}{\numberline {11.2}\textcolor {Chapter }{ Cayley graphs }}{168}{section.11.2}%
\contentsline {subsection}{\numberline {11.2.1}\textcolor {Chapter }{RightCayleyDigraph}}{168}{subsection.11.2.1}%
\contentsline {section}{\numberline {11.3}\textcolor {Chapter }{ Random elements of a semigroup }}{168}{section.11.3}%
\contentsline {subsection}{\numberline {11.3.1}\textcolor {Chapter }{Random (for a semigroup)}}{168}{subsection.11.3.1}%
\contentsline {section}{\numberline {11.4}\textcolor {Chapter }{ Properties of elements in a semigroup }}{169}{section.11.4}%
\contentsline {subsection}{\numberline {11.4.1}\textcolor {Chapter }{IndexPeriodOfSemigroupElement}}{169}{subsection.11.4.1}%
\contentsline {subsection}{\numberline {11.4.2}\textcolor {Chapter }{SmallestIdempotentPower}}{169}{subsection.11.4.2}%
\contentsline {section}{\numberline {11.5}\textcolor {Chapter }{ Operations for elements in a semigroup }}{170}{section.11.5}%
\contentsline {subsection}{\numberline {11.5.1}\textcolor {Chapter }{OneInverseOfSemigroupElement}}{170}{subsection.11.5.1}%
\contentsline {section}{\numberline {11.6}\textcolor {Chapter }{ Expressing semigroup elements as words in generators }}{170}{section.11.6}%
\contentsline {subsection}{\numberline {11.6.1}\textcolor {Chapter }{EvaluateWord}}{170}{subsection.11.6.1}%
\contentsline {subsection}{\numberline {11.6.2}\textcolor {Chapter }{Factorization}}{171}{subsection.11.6.2}%
\contentsline {subsection}{\numberline {11.6.3}\textcolor {Chapter }{MinimalFactorization}}{172}{subsection.11.6.3}%
\contentsline {subsection}{\numberline {11.6.4}\textcolor {Chapter }{NonTrivialFactorization}}{173}{subsection.11.6.4}%
\contentsline {section}{\numberline {11.7}\textcolor {Chapter }{ Generating sets }}{174}{section.11.7}%
\contentsline {subsection}{\numberline {11.7.1}\textcolor {Chapter }{Generators}}{174}{subsection.11.7.1}%
\contentsline {subsection}{\numberline {11.7.2}\textcolor {Chapter }{SmallGeneratingSet}}{175}{subsection.11.7.2}%
\contentsline {subsection}{\numberline {11.7.3}\textcolor {Chapter }{IrredundantGeneratingSubset}}{176}{subsection.11.7.3}%
\contentsline {subsection}{\numberline {11.7.4}\textcolor {Chapter }{MinimalSemigroupGeneratingSet}}{177}{subsection.11.7.4}%
\contentsline {subsection}{\numberline {11.7.5}\textcolor {Chapter }{GeneratorsSmallest (for a semigroup)}}{178}{subsection.11.7.5}%
\contentsline {subsection}{\numberline {11.7.6}\textcolor {Chapter }{IndecomposableElements}}{179}{subsection.11.7.6}%
\contentsline {section}{\numberline {11.8}\textcolor {Chapter }{ Minimal ideals and multiplicative zeros }}{179}{section.11.8}%
\contentsline {subsection}{\numberline {11.8.1}\textcolor {Chapter }{MinimalIdeal}}{180}{subsection.11.8.1}%
\contentsline {subsection}{\numberline {11.8.2}\textcolor {Chapter }{RepresentativeOfMinimalIdeal}}{180}{subsection.11.8.2}%
\contentsline {subsection}{\numberline {11.8.3}\textcolor {Chapter }{MultiplicativeZero}}{181}{subsection.11.8.3}%
\contentsline {subsection}{\numberline {11.8.4}\textcolor {Chapter }{UnderlyingSemigroupOfSemigroupWithAdjoinedZero}}{181}{subsection.11.8.4}%
\contentsline {section}{\numberline {11.9}\textcolor {Chapter }{ Group of units and identity elements }}{182}{section.11.9}%
\contentsline {subsection}{\numberline {11.9.1}\textcolor {Chapter }{GroupOfUnits}}{182}{subsection.11.9.1}%
\contentsline {section}{\numberline {11.10}\textcolor {Chapter }{ Idempotents }}{183}{section.11.10}%
\contentsline {subsection}{\numberline {11.10.1}\textcolor {Chapter }{Idempotents}}{183}{subsection.11.10.1}%
\contentsline {subsection}{\numberline {11.10.2}\textcolor {Chapter }{NrIdempotents}}{184}{subsection.11.10.2}%
\contentsline {subsection}{\numberline {11.10.3}\textcolor {Chapter }{IdempotentGeneratedSubsemigroup}}{185}{subsection.11.10.3}%
\contentsline {section}{\numberline {11.11}\textcolor {Chapter }{ Maximal subsemigroups }}{186}{section.11.11}%
\contentsline {subsection}{\numberline {11.11.1}\textcolor {Chapter }{MaximalSubsemigroups (for a finite semigroup)}}{186}{subsection.11.11.1}%
\contentsline {subsection}{\numberline {11.11.2}\textcolor {Chapter }{NrMaximalSubsemigroups}}{188}{subsection.11.11.2}%
\contentsline {subsection}{\numberline {11.11.3}\textcolor {Chapter }{IsMaximalSubsemigroup}}{188}{subsection.11.11.3}%
\contentsline {section}{\numberline {11.12}\textcolor {Chapter }{ Attributes of transformations and transformation semigroups }}{189}{section.11.12}%
\contentsline {subsection}{\numberline {11.12.1}\textcolor {Chapter }{ComponentRepsOfTransformationSemigroup}}{189}{subsection.11.12.1}%
\contentsline {subsection}{\numberline {11.12.2}\textcolor {Chapter }{ComponentsOfTransformationSemigroup}}{189}{subsection.11.12.2}%
\contentsline {subsection}{\numberline {11.12.3}\textcolor {Chapter }{CyclesOfTransformationSemigroup}}{190}{subsection.11.12.3}%
\contentsline {subsection}{\numberline {11.12.4}\textcolor {Chapter }{DigraphOfAction (for a transformation semigroup, list, and action)}}{190}{subsection.11.12.4}%
\contentsline {subsection}{\numberline {11.12.5}\textcolor {Chapter }{DigraphOfActionOnPoints (for a transformation semigroup)}}{191}{subsection.11.12.5}%
\contentsline {subsection}{\numberline {11.12.6}\textcolor {Chapter }{FixedPointsOfTransformationSemigroup (for a transformation semigroup)}}{191}{subsection.11.12.6}%
\contentsline {subsection}{\numberline {11.12.7}\textcolor {Chapter }{IsTransitive (for a transformation semigroup and a set)}}{192}{subsection.11.12.7}%
\contentsline {subsection}{\numberline {11.12.8}\textcolor {Chapter }{SmallestElementSemigroup}}{192}{subsection.11.12.8}%
\contentsline {subsection}{\numberline {11.12.9}\textcolor {Chapter }{CanonicalTransformation}}{192}{subsection.11.12.9}%
\contentsline {subsection}{\numberline {11.12.10}\textcolor {Chapter }{IsConnectedTransformationSemigroup (for a transformation semigroup)}}{193}{subsection.11.12.10}%
\contentsline {section}{\numberline {11.13}\textcolor {Chapter }{ Attributes of partial perm semigroups }}{193}{section.11.13}%
\contentsline {subsection}{\numberline {11.13.1}\textcolor {Chapter }{ComponentRepsOfPartialPermSemigroup}}{193}{subsection.11.13.1}%
\contentsline {subsection}{\numberline {11.13.2}\textcolor {Chapter }{ComponentsOfPartialPermSemigroup}}{194}{subsection.11.13.2}%
\contentsline {subsection}{\numberline {11.13.3}\textcolor {Chapter }{CyclesOfPartialPerm}}{194}{subsection.11.13.3}%
\contentsline {subsection}{\numberline {11.13.4}\textcolor {Chapter }{CyclesOfPartialPermSemigroup}}{194}{subsection.11.13.4}%
\contentsline {section}{\numberline {11.14}\textcolor {Chapter }{ Attributes of Rees (0\texttt {\symbol {45}})matrix semigroups }}{195}{section.11.14}%
\contentsline {subsection}{\numberline {11.14.1}\textcolor {Chapter }{RZMSDigraph}}{195}{subsection.11.14.1}%
\contentsline {subsection}{\numberline {11.14.2}\textcolor {Chapter }{RZMSConnectedComponents}}{195}{subsection.11.14.2}%
\contentsline {section}{\numberline {11.15}\textcolor {Chapter }{ Attributes of inverse semigroups }}{196}{section.11.15}%
\contentsline {subsection}{\numberline {11.15.1}\textcolor {Chapter }{NaturalLeqInverseSemigroup}}{196}{subsection.11.15.1}%
\contentsline {subsection}{\numberline {11.15.2}\textcolor {Chapter }{JoinIrreducibleDClasses}}{196}{subsection.11.15.2}%
\contentsline {subsection}{\numberline {11.15.3}\textcolor {Chapter }{MajorantClosure}}{196}{subsection.11.15.3}%
\contentsline {subsection}{\numberline {11.15.4}\textcolor {Chapter }{Minorants}}{197}{subsection.11.15.4}%
\contentsline {subsection}{\numberline {11.15.5}\textcolor {Chapter }{PrimitiveIdempotents}}{198}{subsection.11.15.5}%
\contentsline {subsection}{\numberline {11.15.6}\textcolor {Chapter }{RightCosetsOfInverseSemigroup}}{198}{subsection.11.15.6}%
\contentsline {subsection}{\numberline {11.15.7}\textcolor {Chapter }{SameMinorantsSubgroup}}{199}{subsection.11.15.7}%
\contentsline {subsection}{\numberline {11.15.8}\textcolor {Chapter }{SmallerDegreePartialPermRepresentation}}{199}{subsection.11.15.8}%
\contentsline {subsection}{\numberline {11.15.9}\textcolor {Chapter }{VagnerPrestonRepresentation}}{200}{subsection.11.15.9}%
\contentsline {subsection}{\numberline {11.15.10}\textcolor {Chapter }{CharacterTableOfInverseSemigroup}}{201}{subsection.11.15.10}%
\contentsline {subsection}{\numberline {11.15.11}\textcolor {Chapter }{EUnitaryInverseCover}}{202}{subsection.11.15.11}%
\contentsline {section}{\numberline {11.16}\textcolor {Chapter }{ Nambooripad partial order }}{202}{section.11.16}%
\contentsline {subsection}{\numberline {11.16.1}\textcolor {Chapter }{NambooripadLeqRegularSemigroup}}{202}{subsection.11.16.1}%
\contentsline {subsection}{\numberline {11.16.2}\textcolor {Chapter }{NambooripadPartialOrder}}{203}{subsection.11.16.2}%
\contentsline {chapter}{\numberline {12}\textcolor {Chapter }{ Properties of semigroups }}{204}{chapter.12}%
\contentsline {section}{\numberline {12.1}\textcolor {Chapter }{ Arbitrary semigroups }}{204}{section.12.1}%
\contentsline {subsection}{\numberline {12.1.1}\textcolor {Chapter }{IsBand}}{204}{subsection.12.1.1}%
\contentsline {subsection}{\numberline {12.1.2}\textcolor {Chapter }{IsBlockGroup}}{205}{subsection.12.1.2}%
\contentsline {subsection}{\numberline {12.1.3}\textcolor {Chapter }{IsCommutativeSemigroup}}{205}{subsection.12.1.3}%
\contentsline {subsection}{\numberline {12.1.4}\textcolor {Chapter }{IsCompletelyRegularSemigroup}}{206}{subsection.12.1.4}%
\contentsline {subsection}{\numberline {12.1.5}\textcolor {Chapter }{IsCongruenceFreeSemigroup}}{206}{subsection.12.1.5}%
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\contentsline {subsection}{\numberline {12.1.8}\textcolor {Chapter }{IsIdempotentGenerated}}{208}{subsection.12.1.8}%
\contentsline {subsection}{\numberline {12.1.9}\textcolor {Chapter }{IsLeftSimple}}{208}{subsection.12.1.9}%
\contentsline {subsection}{\numberline {12.1.10}\textcolor {Chapter }{IsLeftZeroSemigroup}}{209}{subsection.12.1.10}%
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\contentsline {subsection}{\numberline {12.1.13}\textcolor {Chapter }{IsMonoidAsSemigroup}}{211}{subsection.12.1.13}%
\contentsline {subsection}{\numberline {12.1.14}\textcolor {Chapter }{IsOrthodoxSemigroup}}{211}{subsection.12.1.14}%
\contentsline {subsection}{\numberline {12.1.15}\textcolor {Chapter }{IsRectangularBand}}{212}{subsection.12.1.15}%
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\contentsline {subsection}{\numberline {12.1.17}\textcolor {Chapter }{IsRegularSemigroup}}{213}{subsection.12.1.17}%
\contentsline {subsection}{\numberline {12.1.18}\textcolor {Chapter }{IsRightZeroSemigroup}}{213}{subsection.12.1.18}%
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\contentsline {subsection}{\numberline {12.1.21}\textcolor {Chapter }{IsSemilattice}}{214}{subsection.12.1.21}%
\contentsline {subsection}{\numberline {12.1.22}\textcolor {Chapter }{IsSimpleSemigroup}}{215}{subsection.12.1.22}%
\contentsline {subsection}{\numberline {12.1.23}\textcolor {Chapter }{IsSynchronizingSemigroup (for a transformation semigroup)}}{215}{subsection.12.1.23}%
\contentsline {subsection}{\numberline {12.1.24}\textcolor {Chapter }{IsUnitRegularMonoid}}{216}{subsection.12.1.24}%
\contentsline {subsection}{\numberline {12.1.25}\textcolor {Chapter }{IsZeroGroup}}{216}{subsection.12.1.25}%
\contentsline {subsection}{\numberline {12.1.26}\textcolor {Chapter }{IsZeroRectangularBand}}{216}{subsection.12.1.26}%
\contentsline {subsection}{\numberline {12.1.27}\textcolor {Chapter }{IsZeroSemigroup}}{217}{subsection.12.1.27}%
\contentsline {subsection}{\numberline {12.1.28}\textcolor {Chapter }{IsZeroSimpleSemigroup}}{217}{subsection.12.1.28}%
\contentsline {subsection}{\numberline {12.1.29}\textcolor {Chapter }{IsSelfDualSemigroup}}{218}{subsection.12.1.29}%
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\contentsline {subsection}{\numberline {12.2.1}\textcolor {Chapter }{IsCliffordSemigroup}}{218}{subsection.12.2.1}%
\contentsline {subsection}{\numberline {12.2.2}\textcolor {Chapter }{IsBrandtSemigroup}}{219}{subsection.12.2.2}%
\contentsline {subsection}{\numberline {12.2.3}\textcolor {Chapter }{IsEUnitaryInverseSemigroup}}{219}{subsection.12.2.3}%
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\contentsline {subsection}{\numberline {12.2.5}\textcolor {Chapter }{IsFInverseMonoid}}{221}{subsection.12.2.5}%
\contentsline {subsection}{\numberline {12.2.6}\textcolor {Chapter }{IsFactorisableInverseMonoid}}{221}{subsection.12.2.6}%
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\contentsline {subsection}{\numberline {12.2.10}\textcolor {Chapter }{IsMonogenicInverseMonoid}}{223}{subsection.12.2.10}%
\contentsline {chapter}{\numberline {13}\textcolor {Chapter }{Congruences}}{225}{chapter.13}%
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\contentsline {subsection}{\numberline {13.1.2}\textcolor {Chapter }{IsLeftSemigroupCongruence}}{226}{subsection.13.1.2}%
\contentsline {subsection}{\numberline {13.1.3}\textcolor {Chapter }{IsRightSemigroupCongruence}}{227}{subsection.13.1.3}%
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\contentsline {subsection}{\numberline {13.2.2}\textcolor {Chapter }{LeftSemigroupCongruence}}{228}{subsection.13.2.2}%
\contentsline {subsection}{\numberline {13.2.3}\textcolor {Chapter }{RightSemigroupCongruence}}{228}{subsection.13.2.3}%
\contentsline {section}{\numberline {13.3}\textcolor {Chapter }{ Congruence classes }}{229}{section.13.3}%
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\contentsline {subsection}{\numberline {13.3.2}\textcolor {Chapter }{IsLeftCongruenceClass}}{229}{subsection.13.3.2}%
\contentsline {subsection}{\numberline {13.3.3}\textcolor {Chapter }{IsRightCongruenceClass}}{230}{subsection.13.3.3}%
\contentsline {subsection}{\numberline {13.3.4}\textcolor {Chapter }{NonTrivialEquivalenceClasses}}{230}{subsection.13.3.4}%
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\contentsline {subsection}{\numberline {13.4.3}\textcolor {Chapter }{PrincipalCongruencesOfSemigroup (for a semigroup)}}{234}{subsection.13.4.3}%
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\contentsline {subsection}{\numberline {13.6.2}\textcolor {Chapter }{RMSCongruenceByLinkedTriple}}{247}{subsection.13.6.2}%
\contentsline {subsection}{\numberline {13.6.3}\textcolor {Chapter }{IsRMSCongruenceClassByLinkedTriple}}{247}{subsection.13.6.3}%
\contentsline {subsection}{\numberline {13.6.4}\textcolor {Chapter }{RMSCongruenceClassByLinkedTriple}}{248}{subsection.13.6.4}%
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\contentsline {subsection}{\numberline {13.7.2}\textcolor {Chapter }{InverseSemigroupCongruenceByKernelTrace}}{250}{subsection.13.7.2}%
\contentsline {subsection}{\numberline {13.7.3}\textcolor {Chapter }{AsInverseSemigroupCongruenceByKernelTrace}}{250}{subsection.13.7.3}%
\contentsline {subsection}{\numberline {13.7.4}\textcolor {Chapter }{KernelOfSemigroupCongruence}}{251}{subsection.13.7.4}%
\contentsline {subsection}{\numberline {13.7.5}\textcolor {Chapter }{TraceOfSemigroupCongruence}}{251}{subsection.13.7.5}%
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\contentsline {subsection}{\numberline {13.7.7}\textcolor {Chapter }{MinimumGroupCongruence}}{252}{subsection.13.7.7}%
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\contentsline {subsection}{\numberline {13.8.2}\textcolor {Chapter }{CongruenceByWangPair}}{253}{subsection.13.8.2}%
\contentsline {subsection}{\numberline {13.8.3}\textcolor {Chapter }{AsCongruenceByWangPair}}{254}{subsection.13.8.3}%
\contentsline {subsection}{\numberline {13.8.4}\textcolor {Chapter }{GeneratingCongruencesOfLattice}}{254}{subsection.13.8.4}%
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\contentsline {subsection}{\numberline {13.9.2}\textcolor {Chapter }{IsReesCongruenceClass}}{255}{subsection.13.9.2}%
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\contentsline {subsection}{\numberline {13.10.2}\textcolor {Chapter }{IsUniversalSemigroupCongruenceClass}}{256}{subsection.13.10.2}%
\contentsline {subsection}{\numberline {13.10.3}\textcolor {Chapter }{UniversalSemigroupCongruence}}{256}{subsection.13.10.3}%
\contentsline {subsection}{\numberline {13.10.4}\textcolor {Chapter }{TrivialCongruence}}{257}{subsection.13.10.4}%
\contentsline {chapter}{\numberline {14}\textcolor {Chapter }{ Semigroup homomorphisms }}{258}{chapter.14}%
\contentsline {section}{\numberline {14.1}\textcolor {Chapter }{ Homomorphisms of arbitrary semigroups }}{258}{section.14.1}%
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\contentsline {subsection}{\numberline {14.1.2}\textcolor {Chapter }{SemigroupHomomorphismByFunctionNC}}{258}{subsection.14.1.2}%
\contentsline {subsection}{\numberline {14.1.3}\textcolor {Chapter }{IsSemigroupHomomorphismByImages}}{260}{subsection.14.1.3}%
\contentsline {subsection}{\numberline {14.1.4}\textcolor {Chapter }{IsSemigroupHomomorphismByFunction}}{260}{subsection.14.1.4}%
\contentsline {subsection}{\numberline {14.1.5}\textcolor {Chapter }{AsSemigroupHomomorphismByImages (for a semigroup homomorphism by function)}}{260}{subsection.14.1.5}%
\contentsline {subsection}{\numberline {14.1.6}\textcolor {Chapter }{AsSemigroupHomomorphismByFunction (for a semigroup homomorphism by images)}}{261}{subsection.14.1.6}%
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\contentsline {subsection}{\numberline {14.2.2}\textcolor {Chapter }{SmallestMultiplicationTable}}{262}{subsection.14.2.2}%
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\contentsline {subsection}{\numberline {14.2.6}\textcolor {Chapter }{IsomorphismSemigroups}}{264}{subsection.14.2.6}%
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\contentsline {subsection}{\numberline {14.3.1}\textcolor {Chapter }{IsRMSIsoByTriple}}{268}{subsection.14.3.1}%
\contentsline {subsection}{\numberline {14.3.2}\textcolor {Chapter }{RMSIsoByTriple}}{269}{subsection.14.3.2}%
\contentsline {subsection}{\numberline {14.3.3}\textcolor {Chapter }{ELM{\textunderscore }LIST (for IsRMSIsoByTriple)}}{270}{subsection.14.3.3}%
\contentsline {subsection}{\numberline {14.3.4}\textcolor {Chapter }{CompositionMapping2 (for IsRMSIsoByTriple)}}{270}{subsection.14.3.4}%
\contentsline {subsection}{\numberline {14.3.5}\textcolor {Chapter }{ImagesElm (for IsRMSIsoByTriple)}}{271}{subsection.14.3.5}%
\contentsline {subsection}{\numberline {14.3.6}\textcolor {Chapter }{CanonicalReesZeroMatrixSemigroup}}{271}{subsection.14.3.6}%
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\contentsline {subsection}{\numberline {15.1.2}\textcolor {Chapter }{RandomWord (for two integers)}}{273}{subsection.15.1.2}%
\contentsline {subsection}{\numberline {15.1.3}\textcolor {Chapter }{StandardiseWord}}{274}{subsection.15.1.3}%
\contentsline {subsection}{\numberline {15.1.4}\textcolor {Chapter }{StringToWord (for a string)}}{274}{subsection.15.1.4}%
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\contentsline {subsection}{\numberline {15.2.2}\textcolor {Chapter }{ElementOfFpSemigroup}}{275}{subsection.15.2.2}%
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\contentsline {chapter}{\numberline {16}\textcolor {Chapter }{ Visualising semigroups and elements }}{293}{chapter.16}%
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\contentsline {subsection}{\numberline {16.1.1}\textcolor {Chapter }{DotString}}{293}{subsection.16.1.1}%
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\contentsline {subsection}{\numberline {17.1.2}\textcolor {Chapter }{WriteGenerators}}{299}{subsection.17.1.2}%
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\contentsline {subsection}{\numberline {17.2.3}\textcolor {Chapter }{IteratorFromMultiplicationTableFile}}{301}{subsection.17.2.3}%
\contentsline {chapter}{\numberline {18}\textcolor {Chapter }{Translations}}{302}{chapter.18}%
\contentsline {section}{\numberline {18.1}\textcolor {Chapter }{Methods for translations}}{303}{section.18.1}%
\contentsline {subsection}{\numberline {18.1.1}\textcolor {Chapter }{IsXTranslation}}{303}{subsection.18.1.1}%
\contentsline {subsection}{\numberline {18.1.2}\textcolor {Chapter }{IsBitranslation (for IsAssociativeElement and IsMultiplicativeElementWithOne)}}{303}{subsection.18.1.2}%
\contentsline {subsection}{\numberline {18.1.3}\textcolor {Chapter }{IsXTranslationCollection}}{304}{subsection.18.1.3}%
\contentsline {subsection}{\numberline {18.1.4}\textcolor {Chapter }{XPartOfBitranslation}}{304}{subsection.18.1.4}%
\contentsline {subsection}{\numberline {18.1.5}\textcolor {Chapter }{XTranslation}}{304}{subsection.18.1.5}%
\contentsline {subsection}{\numberline {18.1.6}\textcolor {Chapter }{Bitranslation (for IsBitranslationsSemigroup, IsLeftTranslation, IsRightTranslation)}}{305}{subsection.18.1.6}%
\contentsline {subsection}{\numberline {18.1.7}\textcolor {Chapter }{UnderlyingSemigroup}}{305}{subsection.18.1.7}%
\contentsline {subsection}{\numberline {18.1.8}\textcolor {Chapter }{XTranslationsSemigroupOfFamily}}{305}{subsection.18.1.8}%
\contentsline {subsection}{\numberline {18.1.9}\textcolor {Chapter }{TypeXTranslationSemigroupElements}}{306}{subsection.18.1.9}%
\contentsline {subsection}{\numberline {18.1.10}\textcolor {Chapter }{XTranslations}}{306}{subsection.18.1.10}%
\contentsline {subsection}{\numberline {18.1.11}\textcolor {Chapter }{TranslationalHull (for IsSemigroup and CanUseFroidurePin and IsFinite)}}{306}{subsection.18.1.11}%
\contentsline {subsection}{\numberline {18.1.12}\textcolor {Chapter }{NrXTranslations}}{307}{subsection.18.1.12}%
\contentsline {subsection}{\numberline {18.1.13}\textcolor {Chapter }{InnerXTranslations}}{307}{subsection.18.1.13}%
\contentsline {subsection}{\numberline {18.1.14}\textcolor {Chapter }{InnerTranslationalHull (for IsSemigroup and CanUseFroidurePin and IsFinite)}}{307}{subsection.18.1.14}%
\contentsline {subsection}{\numberline {18.1.15}\textcolor {Chapter }{UnderlyingRepresentatives (for IsTranslationsSemigroup)}}{308}{subsection.18.1.15}%
\contentsline {subsection}{\numberline {18.1.16}\textcolor {Chapter }{ImageSetOfTranslation (for IsSemigroupTranslation)}}{308}{subsection.18.1.16}%
\contentsline {chapter}{References}{311}{chapter*.5}%
\contentsline {chapter}{Index}{312}{section*.6}%
