\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-plus and min-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-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}}{86}{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}}{88}{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 }}{93}{chapter.7}%
\contentsline {section}{\numberline {7.1}\textcolor {Chapter }{ Transformation semigroups }}{93}{section.7.1}%
\contentsline {subsection}{\numberline {7.1.1}\textcolor {Chapter }{CatalanMonoid}}{93}{subsection.7.1.1}%
\contentsline {subsection}{\numberline {7.1.2}\textcolor {Chapter }{EndomorphismsPartition}}{93}{subsection.7.1.2}%
\contentsline {subsection}{\numberline {7.1.3}\textcolor {Chapter }{PartialTransformationMonoid}}{94}{subsection.7.1.3}%
\contentsline {subsection}{\numberline {7.1.4}\textcolor {Chapter }{SingularTransformationSemigroup}}{94}{subsection.7.1.4}%
\contentsline {subsection}{\numberline {7.1.5}\textcolor {Chapter }{Semigroups of order-preserving transformations}}{94}{subsection.7.1.5}%
\contentsline {subsection}{\numberline {7.1.6}\textcolor {Chapter }{EndomorphismMonoid (for a digraph)}}{96}{subsection.7.1.6}%
\contentsline {section}{\numberline {7.2}\textcolor {Chapter }{ Semigroups of partial permutations }}{96}{section.7.2}%
\contentsline {subsection}{\numberline {7.2.1}\textcolor {Chapter }{MunnSemigroup}}{97}{subsection.7.2.1}%
\contentsline {subsection}{\numberline {7.2.2}\textcolor {Chapter }{RookMonoid}}{97}{subsection.7.2.2}%
\contentsline {subsection}{\numberline {7.2.3}\textcolor {Chapter }{Inverse monoids of order-preserving partial permutations}}{97}{subsection.7.2.3}%
\contentsline {section}{\numberline {7.3}\textcolor {Chapter }{ Semigroups of bipartitions }}{98}{section.7.3}%
\contentsline {subsection}{\numberline {7.3.1}\textcolor {Chapter }{PartitionMonoid}}{98}{subsection.7.3.1}%
\contentsline {subsection}{\numberline {7.3.2}\textcolor {Chapter }{BrauerMonoid}}{99}{subsection.7.3.2}%
\contentsline {subsection}{\numberline {7.3.3}\textcolor {Chapter }{JonesMonoid}}{100}{subsection.7.3.3}%
\contentsline {subsection}{\numberline {7.3.4}\textcolor {Chapter }{PartialJonesMonoid}}{100}{subsection.7.3.4}%
\contentsline {subsection}{\numberline {7.3.5}\textcolor {Chapter }{AnnularJonesMonoid}}{101}{subsection.7.3.5}%
\contentsline {subsection}{\numberline {7.3.6}\textcolor {Chapter }{MotzkinMonoid}}{101}{subsection.7.3.6}%
\contentsline {subsection}{\numberline {7.3.7}\textcolor {Chapter }{DualSymmetricInverseSemigroup}}{101}{subsection.7.3.7}%
\contentsline {subsection}{\numberline {7.3.8}\textcolor {Chapter }{UniformBlockBijectionMonoid}}{102}{subsection.7.3.8}%
\contentsline {subsection}{\numberline {7.3.9}\textcolor {Chapter }{PlanarPartitionMonoid}}{103}{subsection.7.3.9}%
\contentsline {subsection}{\numberline {7.3.10}\textcolor {Chapter }{ModularPartitionMonoid}}{104}{subsection.7.3.10}%
\contentsline {subsection}{\numberline {7.3.11}\textcolor {Chapter }{ApsisMonoid}}{104}{subsection.7.3.11}%
\contentsline {section}{\numberline {7.4}\textcolor {Chapter }{ Standard PBR semigroups }}{105}{section.7.4}%
\contentsline {subsection}{\numberline {7.4.1}\textcolor {Chapter }{FullPBRMonoid}}{105}{subsection.7.4.1}%
\contentsline {section}{\numberline {7.5}\textcolor {Chapter }{ Semigroups of matrices over a finite field }}{105}{section.7.5}%
\contentsline {subsection}{\numberline {7.5.1}\textcolor {Chapter }{FullMatrixMonoid}}{106}{subsection.7.5.1}%
\contentsline {subsection}{\numberline {7.5.2}\textcolor {Chapter }{SpecialLinearMonoid}}{106}{subsection.7.5.2}%
\contentsline {subsection}{\numberline {7.5.3}\textcolor {Chapter }{IsFullMatrixMonoid}}{106}{subsection.7.5.3}%
\contentsline {section}{\numberline {7.6}\textcolor {Chapter }{ Semigroups of boolean matrices }}{107}{section.7.6}%
\contentsline {subsection}{\numberline {7.6.1}\textcolor {Chapter }{FullBooleanMatMonoid}}{107}{subsection.7.6.1}%
\contentsline {subsection}{\numberline {7.6.2}\textcolor {Chapter }{RegularBooleanMatMonoid}}{107}{subsection.7.6.2}%
\contentsline {subsection}{\numberline {7.6.3}\textcolor {Chapter }{ReflexiveBooleanMatMonoid}}{108}{subsection.7.6.3}%
\contentsline {subsection}{\numberline {7.6.4}\textcolor {Chapter }{HallMonoid}}{108}{subsection.7.6.4}%
\contentsline {subsection}{\numberline {7.6.5}\textcolor {Chapter }{GossipMonoid}}{108}{subsection.7.6.5}%
\contentsline {subsection}{\numberline {7.6.6}\textcolor {Chapter }{TriangularBooleanMatMonoid}}{109}{subsection.7.6.6}%
\contentsline {section}{\numberline {7.7}\textcolor {Chapter }{ Semigroups of matrices over a semiring }}{109}{section.7.7}%
\contentsline {subsection}{\numberline {7.7.1}\textcolor {Chapter }{FullTropicalMaxPlusMonoid}}{109}{subsection.7.7.1}%
\contentsline {subsection}{\numberline {7.7.2}\textcolor {Chapter }{FullTropicalMinPlusMonoid}}{110}{subsection.7.7.2}%
\contentsline {section}{\numberline {7.8}\textcolor {Chapter }{ Examples in various representations }}{110}{section.7.8}%
\contentsline {subsection}{\numberline {7.8.1}\textcolor {Chapter }{TrivialSemigroup}}{110}{subsection.7.8.1}%
\contentsline {subsection}{\numberline {7.8.2}\textcolor {Chapter }{MonogenicSemigroup}}{111}{subsection.7.8.2}%
\contentsline {subsection}{\numberline {7.8.3}\textcolor {Chapter }{RectangularBand}}{111}{subsection.7.8.3}%
\contentsline {subsection}{\numberline {7.8.4}\textcolor {Chapter }{FreeSemilattice}}{112}{subsection.7.8.4}%
\contentsline {subsection}{\numberline {7.8.5}\textcolor {Chapter }{ZeroSemigroup}}{113}{subsection.7.8.5}%
\contentsline {subsection}{\numberline {7.8.6}\textcolor {Chapter }{LeftZeroSemigroup}}{114}{subsection.7.8.6}%
\contentsline {subsection}{\numberline {7.8.7}\textcolor {Chapter }{BrandtSemigroup}}{115}{subsection.7.8.7}%
\contentsline {section}{\numberline {7.9}\textcolor {Chapter }{ Free bands }}{116}{section.7.9}%
\contentsline {subsection}{\numberline {7.9.1}\textcolor {Chapter }{FreeBand (for a given rank)}}{116}{subsection.7.9.1}%
\contentsline {subsection}{\numberline {7.9.2}\textcolor {Chapter }{IsFreeBandCategory}}{117}{subsection.7.9.2}%
\contentsline {subsection}{\numberline {7.9.3}\textcolor {Chapter }{IsFreeBand (for a given semigroup)}}{117}{subsection.7.9.3}%
\contentsline {subsection}{\numberline {7.9.4}\textcolor {Chapter }{IsFreeBandElement}}{117}{subsection.7.9.4}%
\contentsline {subsection}{\numberline {7.9.5}\textcolor {Chapter }{IsFreeBandElementCollection}}{117}{subsection.7.9.5}%
\contentsline {subsection}{\numberline {7.9.6}\textcolor {Chapter }{IsFreeBandSubsemigroup}}{117}{subsection.7.9.6}%
\contentsline {subsection}{\numberline {7.9.7}\textcolor {Chapter }{ContentOfFreeBandElement}}{118}{subsection.7.9.7}%
\contentsline {subsection}{\numberline {7.9.8}\textcolor {Chapter }{EqualInFreeBand}}{118}{subsection.7.9.8}%
\contentsline {subsection}{\numberline {7.9.9}\textcolor {Chapter }{GreensDClassOfElement (for a free band and element)}}{119}{subsection.7.9.9}%
\contentsline {subsection}{\numberline {7.9.10}\textcolor {Chapter }{Operators}}{119}{subsection.7.9.10}%
\contentsline {section}{\numberline {7.10}\textcolor {Chapter }{ Graph inverse semigroups }}{119}{section.7.10}%
\contentsline {subsection}{\numberline {7.10.1}\textcolor {Chapter }{GraphInverseSemigroup}}{120}{subsection.7.10.1}%
\contentsline {subsection}{\numberline {7.10.2}\textcolor {Chapter }{Range (for a graph inverse semigroup element)}}{121}{subsection.7.10.2}%
\contentsline {subsection}{\numberline {7.10.3}\textcolor {Chapter }{IsVertex (for a graph inverse semigroup element)}}{121}{subsection.7.10.3}%
\contentsline {subsection}{\numberline {7.10.4}\textcolor {Chapter }{IsGraphInverseSemigroup}}{121}{subsection.7.10.4}%
\contentsline {subsection}{\numberline {7.10.5}\textcolor {Chapter }{GraphOfGraphInverseSemigroup}}{122}{subsection.7.10.5}%
\contentsline {subsection}{\numberline {7.10.6}\textcolor {Chapter }{IsGraphInverseSemigroupElementCollection}}{122}{subsection.7.10.6}%
\contentsline {subsection}{\numberline {7.10.7}\textcolor {Chapter }{IsGraphInverseSubsemigroup}}{122}{subsection.7.10.7}%
\contentsline {subsection}{\numberline {7.10.8}\textcolor {Chapter }{VerticesOfGraphInverseSemigroup}}{123}{subsection.7.10.8}%
\contentsline {subsection}{\numberline {7.10.9}\textcolor {Chapter }{IndexOfVertexOfGraphInverseSemigroup}}{123}{subsection.7.10.9}%
\contentsline {section}{\numberline {7.11}\textcolor {Chapter }{ Free inverse semigroups }}{123}{section.7.11}%
\contentsline {subsection}{\numberline {7.11.1}\textcolor {Chapter }{FreeInverseSemigroup (for a given rank)}}{124}{subsection.7.11.1}%
\contentsline {subsection}{\numberline {7.11.2}\textcolor {Chapter }{IsFreeInverseSemigroupCategory}}{125}{subsection.7.11.2}%
\contentsline {subsection}{\numberline {7.11.3}\textcolor {Chapter }{IsFreeInverseSemigroup}}{125}{subsection.7.11.3}%
\contentsline {subsection}{\numberline {7.11.4}\textcolor {Chapter }{IsFreeInverseSemigroupElement}}{125}{subsection.7.11.4}%
\contentsline {subsection}{\numberline {7.11.5}\textcolor {Chapter }{IsFreeInverseSemigroupElementCollection}}{125}{subsection.7.11.5}%
\contentsline {subsection}{\numberline {7.11.6}\textcolor {Chapter }{CanonicalForm (for a free inverse semigroup element)}}{125}{subsection.7.11.6}%
\contentsline {subsection}{\numberline {7.11.7}\textcolor {Chapter }{MinimalWord (for free inverse semigroup element)}}{126}{subsection.7.11.7}%
\contentsline {subsection}{\numberline {7.11.8}\textcolor {Chapter }{Displaying free inverse semigroup elements }}{126}{subsection.7.11.8}%
\contentsline {subsection}{\numberline {7.11.9}\textcolor {Chapter }{Operators for free inverse semigroup elements }}{127}{subsection.7.11.9}%
\contentsline {chapter}{\numberline {8}\textcolor {Chapter }{ Standard constructions }}{128}{chapter.8}%
\contentsline {section}{\numberline {8.1}\textcolor {Chapter }{ Products of semigroups }}{128}{section.8.1}%
\contentsline {subsection}{\numberline {8.1.1}\textcolor {Chapter }{DirectProduct}}{128}{subsection.8.1.1}%
\contentsline {subsection}{\numberline {8.1.2}\textcolor {Chapter }{WreathProduct}}{129}{subsection.8.1.2}%
\contentsline {section}{\numberline {8.2}\textcolor {Chapter }{ Dual semigroups }}{129}{section.8.2}%
\contentsline {subsection}{\numberline {8.2.1}\textcolor {Chapter }{DualSemigroup}}{129}{subsection.8.2.1}%
\contentsline {subsection}{\numberline {8.2.2}\textcolor {Chapter }{IsDualSemigroupRep}}{130}{subsection.8.2.2}%
\contentsline {subsection}{\numberline {8.2.3}\textcolor {Chapter }{IsDualSemigroupElement}}{130}{subsection.8.2.3}%
\contentsline {subsection}{\numberline {8.2.4}\textcolor {Chapter }{AntiIsomorphismDualSemigroup}}{131}{subsection.8.2.4}%
\contentsline {section}{\numberline {8.3}\textcolor {Chapter }{ Strong semilattices of semigroups }}{131}{section.8.3}%
\contentsline {subsection}{\numberline {8.3.1}\textcolor {Chapter }{StrongSemilatticeOfSemigroups}}{132}{subsection.8.3.1}%
\contentsline {subsection}{\numberline {8.3.2}\textcolor {Chapter }{SSSE}}{132}{subsection.8.3.2}%
\contentsline {subsection}{\numberline {8.3.3}\textcolor {Chapter }{IsSSSE}}{133}{subsection.8.3.3}%
\contentsline {subsection}{\numberline {8.3.4}\textcolor {Chapter }{IsStrongSemilatticeOfSemigroups}}{133}{subsection.8.3.4}%
\contentsline {subsection}{\numberline {8.3.5}\textcolor {Chapter }{SemilatticeOfStrongSemilatticeOfSemigroups}}{133}{subsection.8.3.5}%
\contentsline {subsection}{\numberline {8.3.6}\textcolor {Chapter }{SemigroupsOfStrongSemilatticeOfSemigroups}}{133}{subsection.8.3.6}%
\contentsline {subsection}{\numberline {8.3.7}\textcolor {Chapter }{HomomorphismsOfStrongSemilatticeOfSemigroups}}{134}{subsection.8.3.7}%
\contentsline {section}{\numberline {8.4}\textcolor {Chapter }{ McAlister triple semigroups }}{134}{section.8.4}%
\contentsline {subsection}{\numberline {8.4.1}\textcolor {Chapter }{IsMcAlisterTripleSemigroup}}{135}{subsection.8.4.1}%
\contentsline {subsection}{\numberline {8.4.2}\textcolor {Chapter }{McAlisterTripleSemigroup}}{135}{subsection.8.4.2}%
\contentsline {subsection}{\numberline {8.4.3}\textcolor {Chapter }{McAlisterTripleSemigroupGroup}}{136}{subsection.8.4.3}%
\contentsline {subsection}{\numberline {8.4.4}\textcolor {Chapter }{McAlisterTripleSemigroupPartialOrder}}{136}{subsection.8.4.4}%
\contentsline {subsection}{\numberline {8.4.5}\textcolor {Chapter }{McAlisterTripleSemigroupSemilattice}}{136}{subsection.8.4.5}%
\contentsline {subsection}{\numberline {8.4.6}\textcolor {Chapter }{McAlisterTripleSemigroupAction}}{137}{subsection.8.4.6}%
\contentsline {subsection}{\numberline {8.4.7}\textcolor {Chapter }{IsMcAlisterTripleSemigroupElement}}{137}{subsection.8.4.7}%
\contentsline {subsection}{\numberline {8.4.8}\textcolor {Chapter }{McAlisterTripleSemigroupElement}}{137}{subsection.8.4.8}%
\contentsline {chapter}{\numberline {9}\textcolor {Chapter }{ Ideals }}{139}{chapter.9}%
\contentsline {section}{\numberline {9.1}\textcolor {Chapter }{ Creating ideals }}{139}{section.9.1}%
\contentsline {subsection}{\numberline {9.1.1}\textcolor {Chapter }{SemigroupIdeal}}{139}{subsection.9.1.1}%
\contentsline {subsection}{\numberline {9.1.2}\textcolor {Chapter }{Ideals (for a semigroup)}}{140}{subsection.9.1.2}%
\contentsline {section}{\numberline {9.2}\textcolor {Chapter }{ Attributes of ideals }}{140}{section.9.2}%
\contentsline {subsection}{\numberline {9.2.1}\textcolor {Chapter }{GeneratorsOfSemigroupIdeal}}{140}{subsection.9.2.1}%
\contentsline {subsection}{\numberline {9.2.2}\textcolor {Chapter }{MinimalIdealGeneratingSet}}{141}{subsection.9.2.2}%
\contentsline {subsection}{\numberline {9.2.3}\textcolor {Chapter }{SupersemigroupOfIdeal}}{141}{subsection.9.2.3}%
\contentsline {chapter}{\numberline {10}\textcolor {Chapter }{ Green's relations }}{143}{chapter.10}%
\contentsline {section}{\numberline {10.1}\textcolor {Chapter }{ Creating Green's classes and representatives }}{143}{section.10.1}%
\contentsline {subsection}{\numberline {10.1.1}\textcolor {Chapter }{XClassOfYClass}}{143}{subsection.10.1.1}%
\contentsline {subsection}{\numberline {10.1.2}\textcolor {Chapter }{GreensXClassOfElement}}{144}{subsection.10.1.2}%
\contentsline {subsection}{\numberline {10.1.3}\textcolor {Chapter }{GreensXClassOfElementNC}}{145}{subsection.10.1.3}%
\contentsline {subsection}{\numberline {10.1.4}\textcolor {Chapter }{GreensXClasses}}{146}{subsection.10.1.4}%
\contentsline {subsection}{\numberline {10.1.5}\textcolor {Chapter }{XClassReps}}{148}{subsection.10.1.5}%
\contentsline {subsection}{\numberline {10.1.6}\textcolor {Chapter }{MinimalDClass}}{149}{subsection.10.1.6}%
\contentsline {subsection}{\numberline {10.1.7}\textcolor {Chapter }{MaximalXClasses}}{149}{subsection.10.1.7}%
\contentsline {subsection}{\numberline {10.1.8}\textcolor {Chapter }{NrRegularDClasses}}{150}{subsection.10.1.8}%
\contentsline {subsection}{\numberline {10.1.9}\textcolor {Chapter }{NrXClasses}}{150}{subsection.10.1.9}%
\contentsline {subsection}{\numberline {10.1.10}\textcolor {Chapter }{PartialOrderOfXClasses}}{151}{subsection.10.1.10}%
\contentsline {subsection}{\numberline {10.1.11}\textcolor {Chapter }{LengthOfLongestDClassChain}}{152}{subsection.10.1.11}%
\contentsline {subsection}{\numberline {10.1.12}\textcolor {Chapter }{IsGreensDGreaterThanFunc}}{153}{subsection.10.1.12}%
\contentsline {section}{\numberline {10.2}\textcolor {Chapter }{ Iterators and enumerators of classes and representatives }}{154}{section.10.2}%
\contentsline {subsection}{\numberline {10.2.1}\textcolor {Chapter }{IteratorOfXClassReps}}{154}{subsection.10.2.1}%
\contentsline {subsection}{\numberline {10.2.2}\textcolor {Chapter }{IteratorOfXClasses}}{154}{subsection.10.2.2}%
\contentsline {section}{\numberline {10.3}\textcolor {Chapter }{ Properties of Green's classes }}{155}{section.10.3}%
\contentsline {subsection}{\numberline {10.3.1}\textcolor {Chapter }{Less than for Green's classes}}{156}{subsection.10.3.1}%
\contentsline {subsection}{\numberline {10.3.2}\textcolor {Chapter }{IsRegularGreensClass}}{156}{subsection.10.3.2}%
\contentsline {subsection}{\numberline {10.3.3}\textcolor {Chapter }{IsGreensClassNC}}{157}{subsection.10.3.3}%
\contentsline {section}{\numberline {10.4}\textcolor {Chapter }{ Attributes of Green's classes }}{157}{section.10.4}%
\contentsline {subsection}{\numberline {10.4.1}\textcolor {Chapter }{GroupHClass}}{157}{subsection.10.4.1}%
\contentsline {subsection}{\numberline {10.4.2}\textcolor {Chapter }{SchutzenbergerGroup}}{158}{subsection.10.4.2}%
\contentsline {subsection}{\numberline {10.4.3}\textcolor {Chapter }{StructureDescriptionSchutzenbergerGroups}}{159}{subsection.10.4.3}%
\contentsline {subsection}{\numberline {10.4.4}\textcolor {Chapter }{StructureDescriptionMaximalSubgroups}}{160}{subsection.10.4.4}%
\contentsline {subsection}{\numberline {10.4.5}\textcolor {Chapter }{MultiplicativeNeutralElement (for an H-class)}}{160}{subsection.10.4.5}%
\contentsline {subsection}{\numberline {10.4.6}\textcolor {Chapter }{StructureDescription (for an H-class)}}{161}{subsection.10.4.6}%
\contentsline {subsection}{\numberline {10.4.7}\textcolor {Chapter }{InjectionPrincipalFactor}}{161}{subsection.10.4.7}%
\contentsline {subsection}{\numberline {10.4.8}\textcolor {Chapter }{PrincipalFactor}}{162}{subsection.10.4.8}%
\contentsline {section}{\numberline {10.5}\textcolor {Chapter }{ Operations for Green's relations and classes }}{163}{section.10.5}%
\contentsline {subsection}{\numberline {10.5.1}\textcolor {Chapter }{LeftGreensMultiplier}}{163}{subsection.10.5.1}%
\contentsline {chapter}{\numberline {11}\textcolor {Chapter }{ Attributes and operations for semigroups }}{165}{chapter.11}%
\contentsline {section}{\numberline {11.1}\textcolor {Chapter }{ Accessing the elements of a semigroup }}{165}{section.11.1}%
\contentsline {subsection}{\numberline {11.1.1}\textcolor {Chapter }{AsListCanonical}}{165}{subsection.11.1.1}%
\contentsline {subsection}{\numberline {11.1.2}\textcolor {Chapter }{PositionCanonical}}{166}{subsection.11.1.2}%
\contentsline {subsection}{\numberline {11.1.3}\textcolor {Chapter }{Enumerate}}{166}{subsection.11.1.3}%
\contentsline {subsection}{\numberline {11.1.4}\textcolor {Chapter }{IsEnumerated}}{167}{subsection.11.1.4}%
\contentsline {section}{\numberline {11.2}\textcolor {Chapter }{ Cayley graphs }}{167}{section.11.2}%
\contentsline {subsection}{\numberline {11.2.1}\textcolor {Chapter }{RightCayleyDigraph}}{167}{subsection.11.2.1}%
\contentsline {section}{\numberline {11.3}\textcolor {Chapter }{ Random elements of a semigroup }}{167}{section.11.3}%
\contentsline {subsection}{\numberline {11.3.1}\textcolor {Chapter }{Random (for a semigroup)}}{167}{subsection.11.3.1}%
\contentsline {section}{\numberline {11.4}\textcolor {Chapter }{ Properties of elements in a semigroup }}{168}{section.11.4}%
\contentsline {subsection}{\numberline {11.4.1}\textcolor {Chapter }{IndexPeriodOfSemigroupElement}}{168}{subsection.11.4.1}%
\contentsline {subsection}{\numberline {11.4.2}\textcolor {Chapter }{SmallestIdempotentPower}}{168}{subsection.11.4.2}%
\contentsline {section}{\numberline {11.5}\textcolor {Chapter }{ Expressing semigroup elements as words in generators }}{169}{section.11.5}%
\contentsline {subsection}{\numberline {11.5.1}\textcolor {Chapter }{EvaluateWord}}{169}{subsection.11.5.1}%
\contentsline {subsection}{\numberline {11.5.2}\textcolor {Chapter }{Factorization}}{170}{subsection.11.5.2}%
\contentsline {subsection}{\numberline {11.5.3}\textcolor {Chapter }{MinimalFactorization}}{171}{subsection.11.5.3}%
\contentsline {subsection}{\numberline {11.5.4}\textcolor {Chapter }{NonTrivialFactorization}}{171}{subsection.11.5.4}%
\contentsline {section}{\numberline {11.6}\textcolor {Chapter }{ Generating sets }}{172}{section.11.6}%
\contentsline {subsection}{\numberline {11.6.1}\textcolor {Chapter }{Generators}}{172}{subsection.11.6.1}%
\contentsline {subsection}{\numberline {11.6.2}\textcolor {Chapter }{SmallGeneratingSet}}{173}{subsection.11.6.2}%
\contentsline {subsection}{\numberline {11.6.3}\textcolor {Chapter }{IrredundantGeneratingSubset}}{174}{subsection.11.6.3}%
\contentsline {subsection}{\numberline {11.6.4}\textcolor {Chapter }{MinimalSemigroupGeneratingSet}}{176}{subsection.11.6.4}%
\contentsline {subsection}{\numberline {11.6.5}\textcolor {Chapter }{GeneratorsSmallest (for a semigroup)}}{176}{subsection.11.6.5}%
\contentsline {subsection}{\numberline {11.6.6}\textcolor {Chapter }{IndecomposableElements}}{177}{subsection.11.6.6}%
\contentsline {section}{\numberline {11.7}\textcolor {Chapter }{ Minimal ideals and multiplicative zeros }}{178}{section.11.7}%
\contentsline {subsection}{\numberline {11.7.1}\textcolor {Chapter }{MinimalIdeal}}{178}{subsection.11.7.1}%
\contentsline {subsection}{\numberline {11.7.2}\textcolor {Chapter }{RepresentativeOfMinimalIdeal}}{179}{subsection.11.7.2}%
\contentsline {subsection}{\numberline {11.7.3}\textcolor {Chapter }{MultiplicativeZero}}{179}{subsection.11.7.3}%
\contentsline {subsection}{\numberline {11.7.4}\textcolor {Chapter }{UnderlyingSemigroupOfSemigroupWithAdjoinedZero}}{180}{subsection.11.7.4}%
\contentsline {section}{\numberline {11.8}\textcolor {Chapter }{ Group of units and identity elements }}{181}{section.11.8}%
\contentsline {subsection}{\numberline {11.8.1}\textcolor {Chapter }{GroupOfUnits}}{181}{subsection.11.8.1}%
\contentsline {section}{\numberline {11.9}\textcolor {Chapter }{ Idempotents }}{182}{section.11.9}%
\contentsline {subsection}{\numberline {11.9.1}\textcolor {Chapter }{Idempotents}}{182}{subsection.11.9.1}%
\contentsline {subsection}{\numberline {11.9.2}\textcolor {Chapter }{NrIdempotents}}{183}{subsection.11.9.2}%
\contentsline {subsection}{\numberline {11.9.3}\textcolor {Chapter }{IdempotentGeneratedSubsemigroup}}{184}{subsection.11.9.3}%
\contentsline {section}{\numberline {11.10}\textcolor {Chapter }{ Maximal subsemigroups }}{185}{section.11.10}%
\contentsline {subsection}{\numberline {11.10.1}\textcolor {Chapter }{MaximalSubsemigroups (for a finite semigroup)}}{185}{subsection.11.10.1}%
\contentsline {subsection}{\numberline {11.10.2}\textcolor {Chapter }{NrMaximalSubsemigroups}}{186}{subsection.11.10.2}%
\contentsline {subsection}{\numberline {11.10.3}\textcolor {Chapter }{IsMaximalSubsemigroup}}{187}{subsection.11.10.3}%
\contentsline {section}{\numberline {11.11}\textcolor {Chapter }{ Attributes of transformations and transformation semigroups }}{188}{section.11.11}%
\contentsline {subsection}{\numberline {11.11.1}\textcolor {Chapter }{ComponentRepsOfTransformationSemigroup}}{188}{subsection.11.11.1}%
\contentsline {subsection}{\numberline {11.11.2}\textcolor {Chapter }{ComponentsOfTransformationSemigroup}}{188}{subsection.11.11.2}%
\contentsline {subsection}{\numberline {11.11.3}\textcolor {Chapter }{CyclesOfTransformationSemigroup}}{188}{subsection.11.11.3}%
\contentsline {subsection}{\numberline {11.11.4}\textcolor {Chapter }{DigraphOfAction (for a transformation semigroup, list, and action)}}{189}{subsection.11.11.4}%
\contentsline {subsection}{\numberline {11.11.5}\textcolor {Chapter }{DigraphOfActionOnPoints (for a transformation semigroup)}}{189}{subsection.11.11.5}%
\contentsline {subsection}{\numberline {11.11.6}\textcolor {Chapter }{FixedPointsOfTransformationSemigroup (for a transformation semigroup)}}{190}{subsection.11.11.6}%
\contentsline {subsection}{\numberline {11.11.7}\textcolor {Chapter }{IsTransitive (for a transformation semigroup and a set)}}{190}{subsection.11.11.7}%
\contentsline {subsection}{\numberline {11.11.8}\textcolor {Chapter }{SmallestElementSemigroup}}{191}{subsection.11.11.8}%
\contentsline {subsection}{\numberline {11.11.9}\textcolor {Chapter }{CanonicalTransformation}}{191}{subsection.11.11.9}%
\contentsline {subsection}{\numberline {11.11.10}\textcolor {Chapter }{IsConnectedTransformationSemigroup (for a transformation semigroup)}}{192}{subsection.11.11.10}%
\contentsline {section}{\numberline {11.12}\textcolor {Chapter }{ Attributes of partial perm semigroups }}{192}{section.11.12}%
\contentsline {subsection}{\numberline {11.12.1}\textcolor {Chapter }{ComponentRepsOfPartialPermSemigroup}}{192}{subsection.11.12.1}%
\contentsline {subsection}{\numberline {11.12.2}\textcolor {Chapter }{ComponentsOfPartialPermSemigroup}}{192}{subsection.11.12.2}%
\contentsline {subsection}{\numberline {11.12.3}\textcolor {Chapter }{CyclesOfPartialPerm}}{193}{subsection.11.12.3}%
\contentsline {subsection}{\numberline {11.12.4}\textcolor {Chapter }{CyclesOfPartialPermSemigroup}}{193}{subsection.11.12.4}%
\contentsline {section}{\numberline {11.13}\textcolor {Chapter }{ Attributes of Rees (0-)matrix semigroups }}{193}{section.11.13}%
\contentsline {subsection}{\numberline {11.13.1}\textcolor {Chapter }{RZMSDigraph}}{193}{subsection.11.13.1}%
\contentsline {subsection}{\numberline {11.13.2}\textcolor {Chapter }{RZMSConnectedComponents}}{194}{subsection.11.13.2}%
\contentsline {section}{\numberline {11.14}\textcolor {Chapter }{ Attributes of inverse semigroups }}{194}{section.11.14}%
\contentsline {subsection}{\numberline {11.14.1}\textcolor {Chapter }{NaturalLeqInverseSemigroup}}{194}{subsection.11.14.1}%
\contentsline {subsection}{\numberline {11.14.2}\textcolor {Chapter }{JoinIrreducibleDClasses}}{195}{subsection.11.14.2}%
\contentsline {subsection}{\numberline {11.14.3}\textcolor {Chapter }{MajorantClosure}}{195}{subsection.11.14.3}%
\contentsline {subsection}{\numberline {11.14.4}\textcolor {Chapter }{Minorants}}{196}{subsection.11.14.4}%
\contentsline {subsection}{\numberline {11.14.5}\textcolor {Chapter }{PrimitiveIdempotents}}{196}{subsection.11.14.5}%
\contentsline {subsection}{\numberline {11.14.6}\textcolor {Chapter }{RightCosetsOfInverseSemigroup}}{197}{subsection.11.14.6}%
\contentsline {subsection}{\numberline {11.14.7}\textcolor {Chapter }{SameMinorantsSubgroup}}{197}{subsection.11.14.7}%
\contentsline {subsection}{\numberline {11.14.8}\textcolor {Chapter }{SmallerDegreePartialPermRepresentation}}{198}{subsection.11.14.8}%
\contentsline {subsection}{\numberline {11.14.9}\textcolor {Chapter }{VagnerPrestonRepresentation}}{199}{subsection.11.14.9}%
\contentsline {subsection}{\numberline {11.14.10}\textcolor {Chapter }{CharacterTableOfInverseSemigroup}}{200}{subsection.11.14.10}%
\contentsline {subsection}{\numberline {11.14.11}\textcolor {Chapter }{EUnitaryInverseCover}}{201}{subsection.11.14.11}%
\contentsline {section}{\numberline {11.15}\textcolor {Chapter }{ Nambooripad partial order }}{201}{section.11.15}%
\contentsline {subsection}{\numberline {11.15.1}\textcolor {Chapter }{NambooripadLeqRegularSemigroup}}{201}{subsection.11.15.1}%
\contentsline {subsection}{\numberline {11.15.2}\textcolor {Chapter }{NambooripadPartialOrder}}{201}{subsection.11.15.2}%
\contentsline {chapter}{\numberline {12}\textcolor {Chapter }{ Properties of semigroups }}{203}{chapter.12}%
\contentsline {section}{\numberline {12.1}\textcolor {Chapter }{ Arbitrary semigroups }}{203}{section.12.1}%
\contentsline {subsection}{\numberline {12.1.1}\textcolor {Chapter }{IsBand}}{203}{subsection.12.1.1}%
\contentsline {subsection}{\numberline {12.1.2}\textcolor {Chapter }{IsBlockGroup}}{204}{subsection.12.1.2}%
\contentsline {subsection}{\numberline {12.1.3}\textcolor {Chapter }{IsCommutativeSemigroup}}{204}{subsection.12.1.3}%
\contentsline {subsection}{\numberline {12.1.4}\textcolor {Chapter }{IsCompletelyRegularSemigroup}}{205}{subsection.12.1.4}%
\contentsline {subsection}{\numberline {12.1.5}\textcolor {Chapter }{IsCongruenceFreeSemigroup}}{205}{subsection.12.1.5}%
\contentsline {subsection}{\numberline {12.1.6}\textcolor {Chapter }{IsSurjectiveSemigroup}}{206}{subsection.12.1.6}%
\contentsline {subsection}{\numberline {12.1.7}\textcolor {Chapter }{IsGroupAsSemigroup}}{206}{subsection.12.1.7}%
\contentsline {subsection}{\numberline {12.1.8}\textcolor {Chapter }{IsIdempotentGenerated}}{207}{subsection.12.1.8}%
\contentsline {subsection}{\numberline {12.1.9}\textcolor {Chapter }{IsLeftSimple}}{207}{subsection.12.1.9}%
\contentsline {subsection}{\numberline {12.1.10}\textcolor {Chapter }{IsLeftZeroSemigroup}}{208}{subsection.12.1.10}%
\contentsline {subsection}{\numberline {12.1.11}\textcolor {Chapter }{IsMonogenicSemigroup}}{209}{subsection.12.1.11}%
\contentsline {subsection}{\numberline {12.1.12}\textcolor {Chapter }{IsMonogenicMonoid}}{209}{subsection.12.1.12}%
\contentsline {subsection}{\numberline {12.1.13}\textcolor {Chapter }{IsMonoidAsSemigroup}}{210}{subsection.12.1.13}%
\contentsline {subsection}{\numberline {12.1.14}\textcolor {Chapter }{IsOrthodoxSemigroup}}{210}{subsection.12.1.14}%
\contentsline {subsection}{\numberline {12.1.15}\textcolor {Chapter }{IsRectangularBand}}{211}{subsection.12.1.15}%
\contentsline {subsection}{\numberline {12.1.16}\textcolor {Chapter }{IsRectangularGroup}}{211}{subsection.12.1.16}%
\contentsline {subsection}{\numberline {12.1.17}\textcolor {Chapter }{IsRegularSemigroup}}{212}{subsection.12.1.17}%
\contentsline {subsection}{\numberline {12.1.18}\textcolor {Chapter }{IsRightZeroSemigroup}}{212}{subsection.12.1.18}%
\contentsline {subsection}{\numberline {12.1.19}\textcolor {Chapter }{IsXTrivial}}{212}{subsection.12.1.19}%
\contentsline {subsection}{\numberline {12.1.20}\textcolor {Chapter }{IsSemigroupWithAdjoinedZero}}{213}{subsection.12.1.20}%
\contentsline {subsection}{\numberline {12.1.21}\textcolor {Chapter }{IsSemilattice}}{213}{subsection.12.1.21}%
\contentsline {subsection}{\numberline {12.1.22}\textcolor {Chapter }{IsSimpleSemigroup}}{214}{subsection.12.1.22}%
\contentsline {subsection}{\numberline {12.1.23}\textcolor {Chapter }{IsSynchronizingSemigroup (for a transformation semigroup)}}{214}{subsection.12.1.23}%
\contentsline {subsection}{\numberline {12.1.24}\textcolor {Chapter }{IsUnitRegularMonoid}}{215}{subsection.12.1.24}%
\contentsline {subsection}{\numberline {12.1.25}\textcolor {Chapter }{IsZeroGroup}}{215}{subsection.12.1.25}%
\contentsline {subsection}{\numberline {12.1.26}\textcolor {Chapter }{IsZeroRectangularBand}}{215}{subsection.12.1.26}%
\contentsline {subsection}{\numberline {12.1.27}\textcolor {Chapter }{IsZeroSemigroup}}{216}{subsection.12.1.27}%
\contentsline {subsection}{\numberline {12.1.28}\textcolor {Chapter }{IsZeroSimpleSemigroup}}{216}{subsection.12.1.28}%
\contentsline {subsection}{\numberline {12.1.29}\textcolor {Chapter }{IsSelfDualSemigroup}}{217}{subsection.12.1.29}%
\contentsline {section}{\numberline {12.2}\textcolor {Chapter }{ Inverse semigroups }}{217}{section.12.2}%
\contentsline {subsection}{\numberline {12.2.1}\textcolor {Chapter }{IsCliffordSemigroup}}{217}{subsection.12.2.1}%
\contentsline {subsection}{\numberline {12.2.2}\textcolor {Chapter }{IsBrandtSemigroup}}{218}{subsection.12.2.2}%
\contentsline {subsection}{\numberline {12.2.3}\textcolor {Chapter }{IsEUnitaryInverseSemigroup}}{218}{subsection.12.2.3}%
\contentsline {subsection}{\numberline {12.2.4}\textcolor {Chapter }{IsFInverseSemigroup}}{219}{subsection.12.2.4}%
\contentsline {subsection}{\numberline {12.2.5}\textcolor {Chapter }{IsFInverseMonoid}}{220}{subsection.12.2.5}%
\contentsline {subsection}{\numberline {12.2.6}\textcolor {Chapter }{IsFactorisableInverseMonoid}}{220}{subsection.12.2.6}%
\contentsline {subsection}{\numberline {12.2.7}\textcolor {Chapter }{IsJoinIrreducible}}{220}{subsection.12.2.7}%
\contentsline {subsection}{\numberline {12.2.8}\textcolor {Chapter }{IsMajorantlyClosed}}{221}{subsection.12.2.8}%
\contentsline {subsection}{\numberline {12.2.9}\textcolor {Chapter }{IsMonogenicInverseSemigroup}}{222}{subsection.12.2.9}%
\contentsline {subsection}{\numberline {12.2.10}\textcolor {Chapter }{IsMonogenicInverseMonoid}}{222}{subsection.12.2.10}%
\contentsline {chapter}{\numberline {13}\textcolor {Chapter }{Congruences}}{224}{chapter.13}%
\contentsline {section}{\numberline {13.1}\textcolor {Chapter }{Semigroup congruence objects}}{224}{section.13.1}%
\contentsline {subsection}{\numberline {13.1.1}\textcolor {Chapter }{IsSemigroupCongruence}}{224}{subsection.13.1.1}%
\contentsline {subsection}{\numberline {13.1.2}\textcolor {Chapter }{IsLeftSemigroupCongruence}}{225}{subsection.13.1.2}%
\contentsline {subsection}{\numberline {13.1.3}\textcolor {Chapter }{IsRightSemigroupCongruence}}{226}{subsection.13.1.3}%
\contentsline {section}{\numberline {13.2}\textcolor {Chapter }{ Creating congruences }}{226}{section.13.2}%
\contentsline {subsection}{\numberline {13.2.1}\textcolor {Chapter }{SemigroupCongruence}}{226}{subsection.13.2.1}%
\contentsline {subsection}{\numberline {13.2.2}\textcolor {Chapter }{LeftSemigroupCongruence}}{227}{subsection.13.2.2}%
\contentsline {subsection}{\numberline {13.2.3}\textcolor {Chapter }{RightSemigroupCongruence}}{227}{subsection.13.2.3}%
\contentsline {section}{\numberline {13.3}\textcolor {Chapter }{ Congruence classes }}{228}{section.13.3}%
\contentsline {subsection}{\numberline {13.3.1}\textcolor {Chapter }{IsCongruenceClass}}{228}{subsection.13.3.1}%
\contentsline {subsection}{\numberline {13.3.2}\textcolor {Chapter }{IsLeftCongruenceClass}}{228}{subsection.13.3.2}%
\contentsline {subsection}{\numberline {13.3.3}\textcolor {Chapter }{IsRightCongruenceClass}}{229}{subsection.13.3.3}%
\contentsline {subsection}{\numberline {13.3.4}\textcolor {Chapter }{NonTrivialEquivalenceClasses}}{229}{subsection.13.3.4}%
\contentsline {subsection}{\numberline {13.3.5}\textcolor {Chapter }{EquivalenceRelationLookup (for an equivalence relation over a finite semigroup)}}{229}{subsection.13.3.5}%
\contentsline {subsection}{\numberline {13.3.6}\textcolor {Chapter }{EquivalenceRelationCanonicalLookup (for an equivalence relation over a finite semigroup)}}{230}{subsection.13.3.6}%
\contentsline {subsection}{\numberline {13.3.7}\textcolor {Chapter }{EquivalenceRelationCanonicalPartition}}{231}{subsection.13.3.7}%
\contentsline {subsection}{\numberline {13.3.8}\textcolor {Chapter }{OnLeftCongruenceClasses}}{231}{subsection.13.3.8}%
\contentsline {subsection}{\numberline {13.3.9}\textcolor {Chapter }{OnRightCongruenceClasses}}{232}{subsection.13.3.9}%
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\contentsline {subsection}{\numberline {13.4.1}\textcolor {Chapter }{CongruencesOfSemigroup (for a semigroup)}}{232}{subsection.13.4.1}%
\contentsline {subsection}{\numberline {13.4.2}\textcolor {Chapter }{MinimalCongruencesOfSemigroup (for a semigroup)}}{233}{subsection.13.4.2}%
\contentsline {subsection}{\numberline {13.4.3}\textcolor {Chapter }{PrincipalCongruencesOfSemigroup (for a semigroup)}}{233}{subsection.13.4.3}%
\contentsline {subsection}{\numberline {13.4.4}\textcolor {Chapter }{IsCongruencePoset}}{234}{subsection.13.4.4}%
\contentsline {subsection}{\numberline {13.4.5}\textcolor {Chapter }{LatticeOfCongruences (for a semigroup)}}{235}{subsection.13.4.5}%
\contentsline {subsection}{\numberline {13.4.6}\textcolor {Chapter }{CayleyDigraphOfCongruences (for a semigroup)}}{236}{subsection.13.4.6}%
\contentsline {subsection}{\numberline {13.4.7}\textcolor {Chapter }{PosetOfPrincipalCongruences (for a semigroup)}}{237}{subsection.13.4.7}%
\contentsline {subsection}{\numberline {13.4.8}\textcolor {Chapter }{CongruencesOfPoset}}{238}{subsection.13.4.8}%
\contentsline {subsection}{\numberline {13.4.9}\textcolor {Chapter }{UnderlyingSemigroupOfCongruencePoset}}{238}{subsection.13.4.9}%
\contentsline {subsection}{\numberline {13.4.10}\textcolor {Chapter }{PosetOfCongruences}}{239}{subsection.13.4.10}%
\contentsline {subsection}{\numberline {13.4.11}\textcolor {Chapter }{JoinSemilatticeOfCongruences}}{239}{subsection.13.4.11}%
\contentsline {subsection}{\numberline {13.4.12}\textcolor {Chapter }{GeneratingCongruencesOfJoinSemilattice}}{240}{subsection.13.4.12}%
\contentsline {subsection}{\numberline {13.4.13}\textcolor {Chapter }{MinimalCongruences (for a list or collection)}}{240}{subsection.13.4.13}%
\contentsline {subsection}{\numberline {13.4.14}\textcolor {Chapter }{NumberOfRightCongruences (for a semigroup, positive integer, and list or collection)}}{241}{subsection.13.4.14}%
\contentsline {subsection}{\numberline {13.4.15}\textcolor {Chapter }{IteratorOfRightCongruences (for a semigroup, positive integer, and list or collection)}}{242}{subsection.13.4.15}%
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\contentsline {subsection}{\numberline {13.5.1}\textcolor {Chapter }{IsSubrelation}}{243}{subsection.13.5.1}%
\contentsline {subsection}{\numberline {13.5.2}\textcolor {Chapter }{IsSuperrelation}}{243}{subsection.13.5.2}%
\contentsline {subsection}{\numberline {13.5.3}\textcolor {Chapter }{MeetSemigroupCongruences}}{243}{subsection.13.5.3}%
\contentsline {subsection}{\numberline {13.5.4}\textcolor {Chapter }{JoinSemigroupCongruences}}{244}{subsection.13.5.4}%
\contentsline {section}{\numberline {13.6}\textcolor {Chapter }{Congruences on Rees matrix semigroups}}{244}{section.13.6}%
\contentsline {subsection}{\numberline {13.6.1}\textcolor {Chapter }{IsRMSCongruenceByLinkedTriple}}{245}{subsection.13.6.1}%
\contentsline {subsection}{\numberline {13.6.2}\textcolor {Chapter }{RMSCongruenceByLinkedTriple}}{246}{subsection.13.6.2}%
\contentsline {subsection}{\numberline {13.6.3}\textcolor {Chapter }{IsRMSCongruenceClassByLinkedTriple}}{246}{subsection.13.6.3}%
\contentsline {subsection}{\numberline {13.6.4}\textcolor {Chapter }{RMSCongruenceClassByLinkedTriple}}{247}{subsection.13.6.4}%
\contentsline {subsection}{\numberline {13.6.5}\textcolor {Chapter }{IsLinkedTriple}}{247}{subsection.13.6.5}%
\contentsline {subsection}{\numberline {13.6.6}\textcolor {Chapter }{AsSemigroupCongruenceByGeneratingPairs}}{248}{subsection.13.6.6}%
\contentsline {section}{\numberline {13.7}\textcolor {Chapter }{ Congruences on inverse semigroups }}{248}{section.13.7}%
\contentsline {subsection}{\numberline {13.7.1}\textcolor {Chapter }{IsInverseSemigroupCongruenceByKernelTrace}}{248}{subsection.13.7.1}%
\contentsline {subsection}{\numberline {13.7.2}\textcolor {Chapter }{InverseSemigroupCongruenceByKernelTrace}}{249}{subsection.13.7.2}%
\contentsline {subsection}{\numberline {13.7.3}\textcolor {Chapter }{AsInverseSemigroupCongruenceByKernelTrace}}{249}{subsection.13.7.3}%
\contentsline {subsection}{\numberline {13.7.4}\textcolor {Chapter }{KernelOfSemigroupCongruence}}{250}{subsection.13.7.4}%
\contentsline {subsection}{\numberline {13.7.5}\textcolor {Chapter }{TraceOfSemigroupCongruence}}{250}{subsection.13.7.5}%
\contentsline {subsection}{\numberline {13.7.6}\textcolor {Chapter }{IsInverseSemigroupCongruenceClassByKernelTrace}}{251}{subsection.13.7.6}%
\contentsline {subsection}{\numberline {13.7.7}\textcolor {Chapter }{MinimumGroupCongruence}}{251}{subsection.13.7.7}%
\contentsline {section}{\numberline {13.8}\textcolor {Chapter }{ Congruences on graph inverse semigroups }}{251}{section.13.8}%
\contentsline {subsection}{\numberline {13.8.1}\textcolor {Chapter }{IsCongruenceByWangPair}}{251}{subsection.13.8.1}%
\contentsline {subsection}{\numberline {13.8.2}\textcolor {Chapter }{CongruenceByWangPair}}{252}{subsection.13.8.2}%
\contentsline {subsection}{\numberline {13.8.3}\textcolor {Chapter }{AsCongruenceByWangPair}}{253}{subsection.13.8.3}%
\contentsline {subsection}{\numberline {13.8.4}\textcolor {Chapter }{GeneratingCongruencesOfLattice}}{253}{subsection.13.8.4}%
\contentsline {section}{\numberline {13.9}\textcolor {Chapter }{Rees congruences}}{254}{section.13.9}%
\contentsline {subsection}{\numberline {13.9.1}\textcolor {Chapter }{SemigroupIdealOfReesCongruence}}{254}{subsection.13.9.1}%
\contentsline {subsection}{\numberline {13.9.2}\textcolor {Chapter }{IsReesCongruenceClass}}{254}{subsection.13.9.2}%
\contentsline {section}{\numberline {13.10}\textcolor {Chapter }{Universal and trivial congruences}}{255}{section.13.10}%
\contentsline {subsection}{\numberline {13.10.1}\textcolor {Chapter }{IsUniversalSemigroupCongruence}}{255}{subsection.13.10.1}%
\contentsline {subsection}{\numberline {13.10.2}\textcolor {Chapter }{IsUniversalSemigroupCongruenceClass}}{255}{subsection.13.10.2}%
\contentsline {subsection}{\numberline {13.10.3}\textcolor {Chapter }{UniversalSemigroupCongruence}}{255}{subsection.13.10.3}%
\contentsline {subsection}{\numberline {13.10.4}\textcolor {Chapter }{TrivialCongruence}}{256}{subsection.13.10.4}%
\contentsline {chapter}{\numberline {14}\textcolor {Chapter }{ Semigroup homomorphisms }}{257}{chapter.14}%
\contentsline {section}{\numberline {14.1}\textcolor {Chapter }{ Homomorphisms of arbitrary semigroups }}{257}{section.14.1}%
\contentsline {subsection}{\numberline {14.1.1}\textcolor {Chapter }{SemigroupHomomorphismByImages (for two semigroups and two lists)}}{257}{subsection.14.1.1}%
\contentsline {subsection}{\numberline {14.1.2}\textcolor {Chapter }{SemigroupHomomorphismByFunctionNC}}{257}{subsection.14.1.2}%
\contentsline {subsection}{\numberline {14.1.3}\textcolor {Chapter }{IsSemigroupHomomorphismByImages}}{259}{subsection.14.1.3}%
\contentsline {subsection}{\numberline {14.1.4}\textcolor {Chapter }{IsSemigroupHomomorphismByFunction}}{259}{subsection.14.1.4}%
\contentsline {subsection}{\numberline {14.1.5}\textcolor {Chapter }{AsSemigroupHomomorphismByImages (for a semigroup homomorphism by function)}}{259}{subsection.14.1.5}%
\contentsline {subsection}{\numberline {14.1.6}\textcolor {Chapter }{AsSemigroupHomomorphismByFunction (for a semigroup homomorphism by images)}}{260}{subsection.14.1.6}%
\contentsline {subsection}{\numberline {14.1.7}\textcolor {Chapter }{KernelOfSemigroupHomomorphism}}{260}{subsection.14.1.7}%
\contentsline {section}{\numberline {14.2}\textcolor {Chapter }{ Isomorphisms of arbitrary semigroups }}{261}{section.14.2}%
\contentsline {subsection}{\numberline {14.2.1}\textcolor {Chapter }{IsIsomorphicSemigroup}}{261}{subsection.14.2.1}%
\contentsline {subsection}{\numberline {14.2.2}\textcolor {Chapter }{SmallestMultiplicationTable}}{261}{subsection.14.2.2}%
\contentsline {subsection}{\numberline {14.2.3}\textcolor {Chapter }{CanonicalMultiplicationTable}}{262}{subsection.14.2.3}%
\contentsline {subsection}{\numberline {14.2.4}\textcolor {Chapter }{CanonicalMultiplicationTablePerm}}{262}{subsection.14.2.4}%
\contentsline {subsection}{\numberline {14.2.5}\textcolor {Chapter }{OnMultiplicationTable}}{263}{subsection.14.2.5}%
\contentsline {subsection}{\numberline {14.2.6}\textcolor {Chapter }{IsomorphismSemigroups}}{263}{subsection.14.2.6}%
\contentsline {subsection}{\numberline {14.2.7}\textcolor {Chapter }{AutomorphismGroup (for a semigroup)}}{264}{subsection.14.2.7}%
\contentsline {subsection}{\numberline {14.2.8}\textcolor {Chapter }{SemigroupIsomorphismByImages (for two semigroups and two lists)}}{264}{subsection.14.2.8}%
\contentsline {subsection}{\numberline {14.2.9}\textcolor {Chapter }{SemigroupIsomorphismByFunctionNC}}{265}{subsection.14.2.9}%
\contentsline {subsection}{\numberline {14.2.10}\textcolor {Chapter }{IsSemigroupIsomorphismByFunction}}{265}{subsection.14.2.10}%
\contentsline {subsection}{\numberline {14.2.11}\textcolor {Chapter }{AsSemigroupIsomorphismByFunction (for a semigroup homomorphism by images)}}{266}{subsection.14.2.11}%
\contentsline {subsection}{\numberline {14.2.12}\textcolor {Chapter }{SmallerDegreeTransformationRepresentation}}{266}{subsection.14.2.12}%
\contentsline {subsection}{\numberline {14.2.13}\textcolor {Chapter }{MinimalFaithfulTransformationDegree}}{267}{subsection.14.2.13}%
\contentsline {section}{\numberline {14.3}\textcolor {Chapter }{ Isomorphisms of Rees (0-)matrix semigroups }}{267}{section.14.3}%
\contentsline {subsection}{\numberline {14.3.1}\textcolor {Chapter }{IsRMSIsoByTriple}}{267}{subsection.14.3.1}%
\contentsline {subsection}{\numberline {14.3.2}\textcolor {Chapter }{RMSIsoByTriple}}{268}{subsection.14.3.2}%
\contentsline {subsection}{\numberline {14.3.3}\textcolor {Chapter }{ELM{\textunderscore }LIST (for IsRMSIsoByTriple)}}{269}{subsection.14.3.3}%
\contentsline {subsection}{\numberline {14.3.4}\textcolor {Chapter }{CompositionMapping2 (for IsRMSIsoByTriple)}}{269}{subsection.14.3.4}%
\contentsline {subsection}{\numberline {14.3.5}\textcolor {Chapter }{ImagesElm (for IsRMSIsoByTriple)}}{270}{subsection.14.3.5}%
\contentsline {subsection}{\numberline {14.3.6}\textcolor {Chapter }{CanonicalReesZeroMatrixSemigroup}}{270}{subsection.14.3.6}%
\contentsline {subsection}{\numberline {14.3.7}\textcolor {Chapter }{ Operators for isomorphisms of Rees (0-)matrix semigroups }}{271}{subsection.14.3.7}%
\contentsline {chapter}{\numberline {15}\textcolor {Chapter }{ Finitely presented semigroups and Tietze transformations }}{272}{chapter.15}%
\contentsline {section}{\numberline {15.1}\textcolor {Chapter }{ Changing representation for words and strings }}{272}{section.15.1}%
\contentsline {subsection}{\numberline {15.1.1}\textcolor {Chapter }{WordToString (for a string and a list)}}{272}{subsection.15.1.1}%
\contentsline {subsection}{\numberline {15.1.2}\textcolor {Chapter }{RandomWord (for two integers)}}{272}{subsection.15.1.2}%
\contentsline {subsection}{\numberline {15.1.3}\textcolor {Chapter }{StandardiseWord}}{273}{subsection.15.1.3}%
\contentsline {subsection}{\numberline {15.1.4}\textcolor {Chapter }{StringToWord (for a string)}}{273}{subsection.15.1.4}%
\contentsline {section}{\numberline {15.2}\textcolor {Chapter }{ Helper functions }}{274}{section.15.2}%
\contentsline {subsection}{\numberline {15.2.1}\textcolor {Chapter }{ParseRelations}}{274}{subsection.15.2.1}%
\contentsline {subsection}{\numberline {15.2.2}\textcolor {Chapter }{ElementOfFpSemigroup}}{274}{subsection.15.2.2}%
\contentsline {subsection}{\numberline {15.2.3}\textcolor {Chapter }{ElementOfFpMonoid}}{275}{subsection.15.2.3}%
\contentsline {subsection}{\numberline {15.2.4}\textcolor {Chapter }{FreeMonoidAndAssignGeneratorVars}}{275}{subsection.15.2.4}%
\contentsline {subsection}{\numberline {15.2.5}\textcolor {Chapter }{IsSubsemigroupOfFpMonoid}}{275}{subsection.15.2.5}%
\contentsline {section}{\numberline {15.3}\textcolor {Chapter }{ Creating Tietze transformation objects }}{276}{section.15.3}%
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\contentsline {subsection}{\numberline {15.3.2}\textcolor {Chapter }{IsStzPresentation}}{276}{subsection.15.3.2}%
\contentsline {subsection}{\numberline {15.3.3}\textcolor {Chapter }{GeneratorsOfStzPresentation}}{277}{subsection.15.3.3}%
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\contentsline {section}{\numberline {15.4}\textcolor {Chapter }{ Printing Tietze transformation objects }}{279}{section.15.4}%
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\contentsline {subsection}{\numberline {15.4.2}\textcolor {Chapter }{StzPrintRelation}}{279}{subsection.15.4.2}%
\contentsline {subsection}{\numberline {15.4.3}\textcolor {Chapter }{StzPrintGenerators}}{280}{subsection.15.4.3}%
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\contentsline {subsection}{\numberline {15.5.3}\textcolor {Chapter }{StzAddGenerator}}{283}{subsection.15.5.3}%
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\contentsline {subsection}{\numberline {15.5.5}\textcolor {Chapter }{StzSubstituteRelation}}{284}{subsection.15.5.5}%
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\contentsline {subsection}{\numberline {15.6.2}\textcolor {Chapter }{TietzeBackwardMap}}{285}{subsection.15.6.2}%
\contentsline {subsection}{\numberline {15.6.3}\textcolor {Chapter }{StzIsomorphism}}{286}{subsection.15.6.3}%
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\contentsline {subsection}{\numberline {15.8.2}\textcolor {Chapter }{SimplifiedFpSemigroup}}{289}{subsection.15.8.2}%
\contentsline {subsection}{\numberline {15.8.3}\textcolor {Chapter }{UnreducedFpSemigroup (for a semigroup)}}{290}{subsection.15.8.3}%
\contentsline {subsection}{\numberline {15.8.4}\textcolor {Chapter }{FpTietzeIsomorphism}}{290}{subsection.15.8.4}%
\contentsline {chapter}{\numberline {16}\textcolor {Chapter }{ Visualising semigroups and elements }}{292}{chapter.16}%
\contentsline {section}{\numberline {16.1}\textcolor {Chapter }{ \texttt {dot} pictures }}{292}{section.16.1}%
\contentsline {subsection}{\numberline {16.1.1}\textcolor {Chapter }{DotString}}{292}{subsection.16.1.1}%
\contentsline {subsection}{\numberline {16.1.2}\textcolor {Chapter }{DotString (for a Cayley digraph)}}{293}{subsection.16.1.2}%
\contentsline {subsection}{\numberline {16.1.3}\textcolor {Chapter }{DotSemilatticeOfIdempotents}}{294}{subsection.16.1.3}%
\contentsline {subsection}{\numberline {16.1.4}\textcolor {Chapter }{DotLeftCayleyDigraph}}{294}{subsection.16.1.4}%
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\contentsline {subsection}{\numberline {16.2.1}\textcolor {Chapter }{TexString}}{295}{subsection.16.2.1}%
\contentsline {section}{\numberline {16.3}\textcolor {Chapter }{ \texttt {tikz} pictures }}{295}{section.16.3}%
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\contentsline {chapter}{\numberline {17}\textcolor {Chapter }{ IO }}{298}{chapter.17}%
\contentsline {section}{\numberline {17.1}\textcolor {Chapter }{Reading and writing elements to a file}}{298}{section.17.1}%
\contentsline {subsection}{\numberline {17.1.1}\textcolor {Chapter }{ReadGenerators}}{298}{subsection.17.1.1}%
\contentsline {subsection}{\numberline {17.1.2}\textcolor {Chapter }{WriteGenerators}}{298}{subsection.17.1.2}%
\contentsline {subsection}{\numberline {17.1.3}\textcolor {Chapter }{IteratorFromGeneratorsFile}}{299}{subsection.17.1.3}%
\contentsline {section}{\numberline {17.2}\textcolor {Chapter }{Reading and writing multiplication tables to a file}}{299}{section.17.2}%
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\contentsline {chapter}{\numberline {18}\textcolor {Chapter }{Translations}}{301}{chapter.18}%
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\contentsline {subsection}{\numberline {18.1.1}\textcolor {Chapter }{IsXTranslation}}{302}{subsection.18.1.1}%
\contentsline {subsection}{\numberline {18.1.2}\textcolor {Chapter }{IsBitranslation (for IsAssociativeElement and IsMultiplicativeElementWithOne)}}{302}{subsection.18.1.2}%
\contentsline {subsection}{\numberline {18.1.3}\textcolor {Chapter }{IsXTranslationCollection}}{303}{subsection.18.1.3}%
\contentsline {subsection}{\numberline {18.1.4}\textcolor {Chapter }{XPartOfBitranslation}}{303}{subsection.18.1.4}%
\contentsline {subsection}{\numberline {18.1.5}\textcolor {Chapter }{XTranslation}}{303}{subsection.18.1.5}%
\contentsline {subsection}{\numberline {18.1.6}\textcolor {Chapter }{Bitranslation (for IsBitranslationsSemigroup, IsLeftTranslation, IsRightTranslation)}}{304}{subsection.18.1.6}%
\contentsline {subsection}{\numberline {18.1.7}\textcolor {Chapter }{UnderlyingSemigroup}}{304}{subsection.18.1.7}%
\contentsline {subsection}{\numberline {18.1.8}\textcolor {Chapter }{XTranslationsSemigroupOfFamily}}{304}{subsection.18.1.8}%
\contentsline {subsection}{\numberline {18.1.9}\textcolor {Chapter }{TypeXTranslationSemigroupElements}}{305}{subsection.18.1.9}%
\contentsline {subsection}{\numberline {18.1.10}\textcolor {Chapter }{XTranslations}}{305}{subsection.18.1.10}%
\contentsline {subsection}{\numberline {18.1.11}\textcolor {Chapter }{TranslationalHull (for IsSemigroup and CanUseFroidurePin and IsFinite)}}{305}{subsection.18.1.11}%
\contentsline {subsection}{\numberline {18.1.12}\textcolor {Chapter }{NrXTranslations}}{306}{subsection.18.1.12}%
\contentsline {subsection}{\numberline {18.1.13}\textcolor {Chapter }{InnerXTranslations}}{306}{subsection.18.1.13}%
\contentsline {subsection}{\numberline {18.1.14}\textcolor {Chapter }{InnerTranslationalHull (for IsSemigroup and CanUseFroidurePin and IsFinite)}}{306}{subsection.18.1.14}%
\contentsline {subsection}{\numberline {18.1.15}\textcolor {Chapter }{UnderlyingRepresentatives (for IsTranslationsSemigroup)}}{307}{subsection.18.1.15}%
\contentsline {subsection}{\numberline {18.1.16}\textcolor {Chapter }{ImageSetOfTranslation (for IsSemigroupTranslation)}}{307}{subsection.18.1.16}%
\contentsline {chapter}{References}{310}{chapter*.5}%
\contentsline {chapter}{Index}{311}{section*.6}%
