\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\@writefile{toc}{\contentsline {chapter}{\numberline {1}\leavevmode {\color  {Chapter }Introduction}}{5}{chapter.1}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{intro}{{1}{5}{\textcolor {Chapter }{Introduction}}{chapter.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {1.1}\leavevmode {\color  {Chapter }What is the role of the \textsf  {MatricesForHomalg} package in the \textsf  {homalg} project?}}{5}{section.1.1}}
\newlabel{MatricesForHomalg_role}{{1.1}{5}{\textcolor {Chapter }{What is the role of the \textsf {MatricesForHomalg} package in the \textsf {homalg} project?}}{section.1.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.1}\leavevmode {\color  {Chapter }\textsf  {MatricesForHomalg} provides ...}}{5}{subsection.1.1.1}}
\newlabel{MatricesForHomalg-provides}{{1.1.1}{5}{\textcolor {Chapter }{\textsf {MatricesForHomalg} provides ...}}{subsection.1.1.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.2}\leavevmode {\color  {Chapter }\textsf  {homalg} delegates ...}}{5}{subsection.1.1.2}}
\newlabel{homalg-delegates}{{1.1.2}{5}{\textcolor {Chapter }{\textsf {homalg} delegates ...}}{subsection.1.1.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1.3}\leavevmode {\color  {Chapter }The black box concept}}{6}{subsection.1.1.3}}
\newlabel{black box}{{1.1.3}{6}{\textcolor {Chapter }{The black box concept}}{subsection.1.1.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {1.2}\leavevmode {\color  {Chapter }This manual}}{6}{section.1.2}}
\newlabel{overview}{{1.2}{6}{\textcolor {Chapter }{This manual}}{section.1.2}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {2}\leavevmode {\color  {Chapter }Installation of the \textsf  {MatricesForHomalg} Package}}{7}{chapter.2}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{install}{{2}{7}{\textcolor {Chapter }{Installation of the \textsf {MatricesForHomalg} Package}}{chapter.2}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {3}\leavevmode {\color  {Chapter }Rings}}{8}{chapter.3}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{Rings}{{3}{8}{\textcolor {Chapter }{Rings}}{chapter.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.1}\leavevmode {\color  {Chapter }Rings: Category and Representations}}{8}{section.3.1}}
\newlabel{Rings:Category}{{3.1}{8}{\textcolor {Chapter }{Rings: Category and Representations}}{section.3.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1.1}\leavevmode {\color  {Chapter }IsHomalgRing}}{8}{subsection.3.1.1}}
\newlabel{IsHomalgRing}{{3.1.1}{8}{\textcolor {Chapter }{IsHomalgRing}}{subsection.3.1.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1.2}\leavevmode {\color  {Chapter }IsPreHomalgRing}}{8}{subsection.3.1.2}}
\newlabel{IsPreHomalgRing}{{3.1.2}{8}{\textcolor {Chapter }{IsPreHomalgRing}}{subsection.3.1.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1.3}\leavevmode {\color  {Chapter }IsHomalgRingElement}}{8}{subsection.3.1.3}}
\newlabel{IsHomalgRingElement}{{3.1.3}{8}{\textcolor {Chapter }{IsHomalgRingElement}}{subsection.3.1.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1.4}\leavevmode {\color  {Chapter }IsHomalgInternalRingRep}}{9}{subsection.3.1.4}}
\newlabel{IsHomalgInternalRingRep}{{3.1.4}{9}{\textcolor {Chapter }{IsHomalgInternalRingRep}}{subsection.3.1.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.2}\leavevmode {\color  {Chapter }Rings: Constructors}}{9}{section.3.2}}
\newlabel{Rings:Constructors}{{3.2}{9}{\textcolor {Chapter }{Rings: Constructors}}{section.3.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2.1}\leavevmode {\color  {Chapter }HomalgRingOfIntegers (constructor for the integers)}}{9}{subsection.3.2.1}}
\newlabel{HomalgRingOfIntegers:constructor for the integers}{{3.2.1}{9}{\textcolor {Chapter }{HomalgRingOfIntegers (constructor for the integers)}}{subsection.3.2.1}{}}
\newlabel{HomalgRingOfIntegers:constructor for the residue class rings of the integers}{{3.2.1}{9}{\textcolor {Chapter }{HomalgRingOfIntegers (constructor for the integers)}}{subsection.3.2.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2.2}\leavevmode {\color  {Chapter }HomalgFieldOfRationals (constructor for the field of rationals)}}{10}{subsection.3.2.2}}
\newlabel{HomalgFieldOfRationals:constructor for the field of rationals}{{3.2.2}{10}{\textcolor {Chapter }{HomalgFieldOfRationals (constructor for the field of rationals)}}{subsection.3.2.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2.3}\leavevmode {\color  {Chapter }\texttt  {\char 92\relax }/ (constructor for residue class rings)}}{10}{subsection.3.2.3}}
\newlabel{/:constructor for residue class rings}{{3.2.3}{10}{\textcolor {Chapter }{\texttt {\symbol {92}}/ (constructor for residue class rings)}}{subsection.3.2.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.3}\leavevmode {\color  {Chapter }Rings: Properties}}{11}{section.3.3}}
\newlabel{Rings:Properties}{{3.3}{11}{\textcolor {Chapter }{Rings: Properties}}{section.3.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.1}\leavevmode {\color  {Chapter }IsZero (for rings)}}{11}{subsection.3.3.1}}
\newlabel{IsZero:for rings}{{3.3.1}{11}{\textcolor {Chapter }{IsZero (for rings)}}{subsection.3.3.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.2}\leavevmode {\color  {Chapter }ContainsAField}}{11}{subsection.3.3.2}}
\newlabel{ContainsAField}{{3.3.2}{11}{\textcolor {Chapter }{ContainsAField}}{subsection.3.3.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.3}\leavevmode {\color  {Chapter }IsRationalsForHomalg}}{11}{subsection.3.3.3}}
\newlabel{IsRationalsForHomalg}{{3.3.3}{11}{\textcolor {Chapter }{IsRationalsForHomalg}}{subsection.3.3.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.4}\leavevmode {\color  {Chapter }IsFieldForHomalg}}{11}{subsection.3.3.4}}
\newlabel{IsFieldForHomalg}{{3.3.4}{11}{\textcolor {Chapter }{IsFieldForHomalg}}{subsection.3.3.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.5}\leavevmode {\color  {Chapter }IsDivisionRingForHomalg}}{11}{subsection.3.3.5}}
\newlabel{IsDivisionRingForHomalg}{{3.3.5}{11}{\textcolor {Chapter }{IsDivisionRingForHomalg}}{subsection.3.3.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.6}\leavevmode {\color  {Chapter }IsIntegersForHomalg}}{12}{subsection.3.3.6}}
\newlabel{IsIntegersForHomalg}{{3.3.6}{12}{\textcolor {Chapter }{IsIntegersForHomalg}}{subsection.3.3.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.7}\leavevmode {\color  {Chapter }IsResidueClassRingOfTheIntegers}}{12}{subsection.3.3.7}}
\newlabel{IsResidueClassRingOfTheIntegers}{{3.3.7}{12}{\textcolor {Chapter }{IsResidueClassRingOfTheIntegers}}{subsection.3.3.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.8}\leavevmode {\color  {Chapter }IsBezoutRing}}{12}{subsection.3.3.8}}
\newlabel{IsBezoutRing}{{3.3.8}{12}{\textcolor {Chapter }{IsBezoutRing}}{subsection.3.3.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.9}\leavevmode {\color  {Chapter }IsIntegrallyClosedDomain}}{12}{subsection.3.3.9}}
\newlabel{IsIntegrallyClosedDomain}{{3.3.9}{12}{\textcolor {Chapter }{IsIntegrallyClosedDomain}}{subsection.3.3.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.10}\leavevmode {\color  {Chapter }IsUniqueFactorizationDomain}}{12}{subsection.3.3.10}}
\newlabel{IsUniqueFactorizationDomain}{{3.3.10}{12}{\textcolor {Chapter }{IsUniqueFactorizationDomain}}{subsection.3.3.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.11}\leavevmode {\color  {Chapter }IsKaplanskyHermite}}{12}{subsection.3.3.11}}
\newlabel{IsKaplanskyHermite}{{3.3.11}{12}{\textcolor {Chapter }{IsKaplanskyHermite}}{subsection.3.3.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.12}\leavevmode {\color  {Chapter }IsDedekindDomain}}{12}{subsection.3.3.12}}
\newlabel{IsDedekindDomain}{{3.3.12}{12}{\textcolor {Chapter }{IsDedekindDomain}}{subsection.3.3.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.13}\leavevmode {\color  {Chapter }IsDiscreteValuationRing}}{12}{subsection.3.3.13}}
\newlabel{IsDiscreteValuationRing}{{3.3.13}{12}{\textcolor {Chapter }{IsDiscreteValuationRing}}{subsection.3.3.13}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.14}\leavevmode {\color  {Chapter }IsFreePolynomialRing}}{13}{subsection.3.3.14}}
\newlabel{IsFreePolynomialRing}{{3.3.14}{13}{\textcolor {Chapter }{IsFreePolynomialRing}}{subsection.3.3.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.15}\leavevmode {\color  {Chapter }IsWeylRing}}{13}{subsection.3.3.15}}
\newlabel{IsWeylRing}{{3.3.15}{13}{\textcolor {Chapter }{IsWeylRing}}{subsection.3.3.15}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.16}\leavevmode {\color  {Chapter }IsLocalizedWeylRing}}{13}{subsection.3.3.16}}
\newlabel{IsLocalizedWeylRing}{{3.3.16}{13}{\textcolor {Chapter }{IsLocalizedWeylRing}}{subsection.3.3.16}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.17}\leavevmode {\color  {Chapter }IsGlobalDimensionFinite}}{13}{subsection.3.3.17}}
\newlabel{IsGlobalDimensionFinite}{{3.3.17}{13}{\textcolor {Chapter }{IsGlobalDimensionFinite}}{subsection.3.3.17}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.18}\leavevmode {\color  {Chapter }IsLeftGlobalDimensionFinite}}{13}{subsection.3.3.18}}
\newlabel{IsLeftGlobalDimensionFinite}{{3.3.18}{13}{\textcolor {Chapter }{IsLeftGlobalDimensionFinite}}{subsection.3.3.18}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.19}\leavevmode {\color  {Chapter }IsRightGlobalDimensionFinite}}{13}{subsection.3.3.19}}
\newlabel{IsRightGlobalDimensionFinite}{{3.3.19}{13}{\textcolor {Chapter }{IsRightGlobalDimensionFinite}}{subsection.3.3.19}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.20}\leavevmode {\color  {Chapter }HasInvariantBasisProperty}}{13}{subsection.3.3.20}}
\newlabel{HasInvariantBasisProperty}{{3.3.20}{13}{\textcolor {Chapter }{HasInvariantBasisProperty}}{subsection.3.3.20}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.21}\leavevmode {\color  {Chapter }HasLeftInvariantBasisProperty}}{13}{subsection.3.3.21}}
\newlabel{HasLeftInvariantBasisProperty}{{3.3.21}{13}{\textcolor {Chapter }{HasLeftInvariantBasisProperty}}{subsection.3.3.21}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.22}\leavevmode {\color  {Chapter }HasRightInvariantBasisProperty}}{14}{subsection.3.3.22}}
\newlabel{HasRightInvariantBasisProperty}{{3.3.22}{14}{\textcolor {Chapter }{HasRightInvariantBasisProperty}}{subsection.3.3.22}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.23}\leavevmode {\color  {Chapter }IsLocal}}{14}{subsection.3.3.23}}
\newlabel{IsLocal}{{3.3.23}{14}{\textcolor {Chapter }{IsLocal}}{subsection.3.3.23}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.24}\leavevmode {\color  {Chapter }IsSemiLocalRing}}{14}{subsection.3.3.24}}
\newlabel{IsSemiLocalRing}{{3.3.24}{14}{\textcolor {Chapter }{IsSemiLocalRing}}{subsection.3.3.24}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.25}\leavevmode {\color  {Chapter }IsIntegralDomain}}{14}{subsection.3.3.25}}
\newlabel{IsIntegralDomain}{{3.3.25}{14}{\textcolor {Chapter }{IsIntegralDomain}}{subsection.3.3.25}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.26}\leavevmode {\color  {Chapter }IsHereditary}}{14}{subsection.3.3.26}}
\newlabel{IsHereditary}{{3.3.26}{14}{\textcolor {Chapter }{IsHereditary}}{subsection.3.3.26}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.27}\leavevmode {\color  {Chapter }IsLeftHereditary}}{14}{subsection.3.3.27}}
\newlabel{IsLeftHereditary}{{3.3.27}{14}{\textcolor {Chapter }{IsLeftHereditary}}{subsection.3.3.27}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.28}\leavevmode {\color  {Chapter }IsRightHereditary}}{14}{subsection.3.3.28}}
\newlabel{IsRightHereditary}{{3.3.28}{14}{\textcolor {Chapter }{IsRightHereditary}}{subsection.3.3.28}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.29}\leavevmode {\color  {Chapter }IsHermite}}{14}{subsection.3.3.29}}
\newlabel{IsHermite}{{3.3.29}{14}{\textcolor {Chapter }{IsHermite}}{subsection.3.3.29}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.30}\leavevmode {\color  {Chapter }IsLeftHermite}}{15}{subsection.3.3.30}}
\newlabel{IsLeftHermite}{{3.3.30}{15}{\textcolor {Chapter }{IsLeftHermite}}{subsection.3.3.30}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.31}\leavevmode {\color  {Chapter }IsRightHermite}}{15}{subsection.3.3.31}}
\newlabel{IsRightHermite}{{3.3.31}{15}{\textcolor {Chapter }{IsRightHermite}}{subsection.3.3.31}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.32}\leavevmode {\color  {Chapter }IsNoetherian}}{15}{subsection.3.3.32}}
\newlabel{IsNoetherian}{{3.3.32}{15}{\textcolor {Chapter }{IsNoetherian}}{subsection.3.3.32}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.33}\leavevmode {\color  {Chapter }IsLeftNoetherian}}{15}{subsection.3.3.33}}
\newlabel{IsLeftNoetherian}{{3.3.33}{15}{\textcolor {Chapter }{IsLeftNoetherian}}{subsection.3.3.33}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.34}\leavevmode {\color  {Chapter }IsRightNoetherian}}{15}{subsection.3.3.34}}
\newlabel{IsRightNoetherian}{{3.3.34}{15}{\textcolor {Chapter }{IsRightNoetherian}}{subsection.3.3.34}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.35}\leavevmode {\color  {Chapter }IsCohenMacaulay}}{15}{subsection.3.3.35}}
\newlabel{IsCohenMacaulay}{{3.3.35}{15}{\textcolor {Chapter }{IsCohenMacaulay}}{subsection.3.3.35}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.36}\leavevmode {\color  {Chapter }IsGorenstein}}{15}{subsection.3.3.36}}
\newlabel{IsGorenstein}{{3.3.36}{15}{\textcolor {Chapter }{IsGorenstein}}{subsection.3.3.36}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.37}\leavevmode {\color  {Chapter }IsKoszul}}{15}{subsection.3.3.37}}
\newlabel{IsKoszul}{{3.3.37}{15}{\textcolor {Chapter }{IsKoszul}}{subsection.3.3.37}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.38}\leavevmode {\color  {Chapter }IsArtinian (for rings)}}{16}{subsection.3.3.38}}
\newlabel{IsArtinian:for rings}{{3.3.38}{16}{\textcolor {Chapter }{IsArtinian (for rings)}}{subsection.3.3.38}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.39}\leavevmode {\color  {Chapter }IsLeftArtinian}}{16}{subsection.3.3.39}}
\newlabel{IsLeftArtinian}{{3.3.39}{16}{\textcolor {Chapter }{IsLeftArtinian}}{subsection.3.3.39}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.40}\leavevmode {\color  {Chapter }IsRightArtinian}}{16}{subsection.3.3.40}}
\newlabel{IsRightArtinian}{{3.3.40}{16}{\textcolor {Chapter }{IsRightArtinian}}{subsection.3.3.40}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.41}\leavevmode {\color  {Chapter }IsOreDomain}}{16}{subsection.3.3.41}}
\newlabel{IsOreDomain}{{3.3.41}{16}{\textcolor {Chapter }{IsOreDomain}}{subsection.3.3.41}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.42}\leavevmode {\color  {Chapter }IsLeftOreDomain}}{16}{subsection.3.3.42}}
\newlabel{IsLeftOreDomain}{{3.3.42}{16}{\textcolor {Chapter }{IsLeftOreDomain}}{subsection.3.3.42}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.43}\leavevmode {\color  {Chapter }IsRightOreDomain}}{16}{subsection.3.3.43}}
\newlabel{IsRightOreDomain}{{3.3.43}{16}{\textcolor {Chapter }{IsRightOreDomain}}{subsection.3.3.43}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.44}\leavevmode {\color  {Chapter }IsPrincipalIdealRing}}{16}{subsection.3.3.44}}
\newlabel{IsPrincipalIdealRing}{{3.3.44}{16}{\textcolor {Chapter }{IsPrincipalIdealRing}}{subsection.3.3.44}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.45}\leavevmode {\color  {Chapter }IsLeftPrincipalIdealRing}}{16}{subsection.3.3.45}}
\newlabel{IsLeftPrincipalIdealRing}{{3.3.45}{16}{\textcolor {Chapter }{IsLeftPrincipalIdealRing}}{subsection.3.3.45}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.46}\leavevmode {\color  {Chapter }IsRightPrincipalIdealRing}}{17}{subsection.3.3.46}}
\newlabel{IsRightPrincipalIdealRing}{{3.3.46}{17}{\textcolor {Chapter }{IsRightPrincipalIdealRing}}{subsection.3.3.46}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.47}\leavevmode {\color  {Chapter }IsRegular}}{17}{subsection.3.3.47}}
\newlabel{IsRegular}{{3.3.47}{17}{\textcolor {Chapter }{IsRegular}}{subsection.3.3.47}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.48}\leavevmode {\color  {Chapter }IsFiniteFreePresentationRing}}{17}{subsection.3.3.48}}
\newlabel{IsFiniteFreePresentationRing}{{3.3.48}{17}{\textcolor {Chapter }{IsFiniteFreePresentationRing}}{subsection.3.3.48}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.49}\leavevmode {\color  {Chapter }IsLeftFiniteFreePresentationRing}}{17}{subsection.3.3.49}}
\newlabel{IsLeftFiniteFreePresentationRing}{{3.3.49}{17}{\textcolor {Chapter }{IsLeftFiniteFreePresentationRing}}{subsection.3.3.49}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.50}\leavevmode {\color  {Chapter }IsRightFiniteFreePresentationRing}}{17}{subsection.3.3.50}}
\newlabel{IsRightFiniteFreePresentationRing}{{3.3.50}{17}{\textcolor {Chapter }{IsRightFiniteFreePresentationRing}}{subsection.3.3.50}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.51}\leavevmode {\color  {Chapter }IsSimpleRing}}{17}{subsection.3.3.51}}
\newlabel{IsSimpleRing}{{3.3.51}{17}{\textcolor {Chapter }{IsSimpleRing}}{subsection.3.3.51}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.52}\leavevmode {\color  {Chapter }IsSemiSimpleRing}}{17}{subsection.3.3.52}}
\newlabel{IsSemiSimpleRing}{{3.3.52}{17}{\textcolor {Chapter }{IsSemiSimpleRing}}{subsection.3.3.52}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.53}\leavevmode {\color  {Chapter }IsSuperCommutative}}{17}{subsection.3.3.53}}
\newlabel{IsSuperCommutative}{{3.3.53}{17}{\textcolor {Chapter }{IsSuperCommutative}}{subsection.3.3.53}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.54}\leavevmode {\color  {Chapter }BasisAlgorithmRespectsPrincipalIdeals}}{18}{subsection.3.3.54}}
\newlabel{BasisAlgorithmRespectsPrincipalIdeals}{{3.3.54}{18}{\textcolor {Chapter }{BasisAlgorithmRespectsPrincipalIdeals}}{subsection.3.3.54}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.55}\leavevmode {\color  {Chapter }AreUnitsCentral}}{18}{subsection.3.3.55}}
\newlabel{AreUnitsCentral}{{3.3.55}{18}{\textcolor {Chapter }{AreUnitsCentral}}{subsection.3.3.55}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.56}\leavevmode {\color  {Chapter }IsMinusOne}}{18}{subsection.3.3.56}}
\newlabel{IsMinusOne}{{3.3.56}{18}{\textcolor {Chapter }{IsMinusOne}}{subsection.3.3.56}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.57}\leavevmode {\color  {Chapter }IsMonic (for homalg ring elements)}}{18}{subsection.3.3.57}}
\newlabel{IsMonic:for homalg ring elements}{{3.3.57}{18}{\textcolor {Chapter }{IsMonic (for homalg ring elements)}}{subsection.3.3.57}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.58}\leavevmode {\color  {Chapter }IsMonicUptoUnit (for homalg ring elements)}}{18}{subsection.3.3.58}}
\newlabel{IsMonicUptoUnit:for homalg ring elements}{{3.3.58}{18}{\textcolor {Chapter }{IsMonicUptoUnit (for homalg ring elements)}}{subsection.3.3.58}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.59}\leavevmode {\color  {Chapter }IsLeftRegular (for homalg ring elements)}}{18}{subsection.3.3.59}}
\newlabel{IsLeftRegular:for homalg ring elements}{{3.3.59}{18}{\textcolor {Chapter }{IsLeftRegular (for homalg ring elements)}}{subsection.3.3.59}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.60}\leavevmode {\color  {Chapter }IsRightRegular (for homalg ring elements)}}{18}{subsection.3.3.60}}
\newlabel{IsRightRegular:for homalg ring elements}{{3.3.60}{18}{\textcolor {Chapter }{IsRightRegular (for homalg ring elements)}}{subsection.3.3.60}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.61}\leavevmode {\color  {Chapter }IsRegular (for homalg ring elements)}}{18}{subsection.3.3.61}}
\newlabel{IsRegular:for homalg ring elements}{{3.3.61}{18}{\textcolor {Chapter }{IsRegular (for homalg ring elements)}}{subsection.3.3.61}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.4}\leavevmode {\color  {Chapter }Rings: Attributes}}{19}{section.3.4}}
\newlabel{Rings:Attributes}{{3.4}{19}{\textcolor {Chapter }{Rings: Attributes}}{section.3.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.1}\leavevmode {\color  {Chapter }Inverse (for homalg ring elements)}}{19}{subsection.3.4.1}}
\newlabel{Inverse:for homalg ring elements}{{3.4.1}{19}{\textcolor {Chapter }{Inverse (for homalg ring elements)}}{subsection.3.4.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.2}\leavevmode {\color  {Chapter }homalgTable}}{19}{subsection.3.4.2}}
\newlabel{homalgTable}{{3.4.2}{19}{\textcolor {Chapter }{homalgTable}}{subsection.3.4.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.3}\leavevmode {\color  {Chapter }RingElementConstructor}}{19}{subsection.3.4.3}}
\newlabel{RingElementConstructor}{{3.4.3}{19}{\textcolor {Chapter }{RingElementConstructor}}{subsection.3.4.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.4}\leavevmode {\color  {Chapter }TypeOfHomalgMatrix}}{19}{subsection.3.4.4}}
\newlabel{TypeOfHomalgMatrix}{{3.4.4}{19}{\textcolor {Chapter }{TypeOfHomalgMatrix}}{subsection.3.4.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.5}\leavevmode {\color  {Chapter }ConstructorForHomalgMatrices}}{19}{subsection.3.4.5}}
\newlabel{ConstructorForHomalgMatrices}{{3.4.5}{19}{\textcolor {Chapter }{ConstructorForHomalgMatrices}}{subsection.3.4.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.6}\leavevmode {\color  {Chapter }Zero (for homalg rings)}}{20}{subsection.3.4.6}}
\newlabel{Zero:for homalg rings}{{3.4.6}{20}{\textcolor {Chapter }{Zero (for homalg rings)}}{subsection.3.4.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.7}\leavevmode {\color  {Chapter }One (for homalg rings)}}{20}{subsection.3.4.7}}
\newlabel{One:for homalg rings}{{3.4.7}{20}{\textcolor {Chapter }{One (for homalg rings)}}{subsection.3.4.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.8}\leavevmode {\color  {Chapter }MinusOne}}{20}{subsection.3.4.8}}
\newlabel{MinusOne}{{3.4.8}{20}{\textcolor {Chapter }{MinusOne}}{subsection.3.4.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.9}\leavevmode {\color  {Chapter }ProductOfIndeterminates}}{20}{subsection.3.4.9}}
\newlabel{ProductOfIndeterminates}{{3.4.9}{20}{\textcolor {Chapter }{ProductOfIndeterminates}}{subsection.3.4.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.10}\leavevmode {\color  {Chapter }RationalParameters}}{20}{subsection.3.4.10}}
\newlabel{RationalParameters}{{3.4.10}{20}{\textcolor {Chapter }{RationalParameters}}{subsection.3.4.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.11}\leavevmode {\color  {Chapter }IndeterminatesOfPolynomialRing}}{20}{subsection.3.4.11}}
\newlabel{IndeterminatesOfPolynomialRing}{{3.4.11}{20}{\textcolor {Chapter }{IndeterminatesOfPolynomialRing}}{subsection.3.4.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.12}\leavevmode {\color  {Chapter }RelativeIndeterminatesOfPolynomialRing}}{20}{subsection.3.4.12}}
\newlabel{RelativeIndeterminatesOfPolynomialRing}{{3.4.12}{20}{\textcolor {Chapter }{RelativeIndeterminatesOfPolynomialRing}}{subsection.3.4.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.13}\leavevmode {\color  {Chapter }IndeterminateCoordinatesOfRingOfDerivations}}{20}{subsection.3.4.13}}
\newlabel{IndeterminateCoordinatesOfRingOfDerivations}{{3.4.13}{20}{\textcolor {Chapter }{IndeterminateCoordinatesOfRingOfDerivations}}{subsection.3.4.13}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.14}\leavevmode {\color  {Chapter }RelativeIndeterminateCoordinatesOfRingOfDerivations}}{21}{subsection.3.4.14}}
\newlabel{RelativeIndeterminateCoordinatesOfRingOfDerivations}{{3.4.14}{21}{\textcolor {Chapter }{RelativeIndeterminateCoordinatesOfRingOfDerivations}}{subsection.3.4.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.15}\leavevmode {\color  {Chapter }IndeterminateDerivationsOfRingOfDerivations}}{21}{subsection.3.4.15}}
\newlabel{IndeterminateDerivationsOfRingOfDerivations}{{3.4.15}{21}{\textcolor {Chapter }{IndeterminateDerivationsOfRingOfDerivations}}{subsection.3.4.15}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.16}\leavevmode {\color  {Chapter }RelativeIndeterminateDerivationsOfRingOfDerivations}}{21}{subsection.3.4.16}}
\newlabel{RelativeIndeterminateDerivationsOfRingOfDerivations}{{3.4.16}{21}{\textcolor {Chapter }{RelativeIndeterminateDerivationsOfRingOfDerivations}}{subsection.3.4.16}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.17}\leavevmode {\color  {Chapter }IndeterminateAntiCommutingVariablesOfExteriorRing}}{21}{subsection.3.4.17}}
\newlabel{IndeterminateAntiCommutingVariablesOfExteriorRing}{{3.4.17}{21}{\textcolor {Chapter }{IndeterminateAntiCommutingVariablesOfExteriorRing}}{subsection.3.4.17}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.18}\leavevmode {\color  {Chapter }RelativeIndeterminateAntiCommutingVariablesOfExteriorRing}}{21}{subsection.3.4.18}}
\newlabel{RelativeIndeterminateAntiCommutingVariablesOfExteriorRing}{{3.4.18}{21}{\textcolor {Chapter }{RelativeIndeterminateAntiCommutingVariablesOfExteriorRing}}{subsection.3.4.18}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.19}\leavevmode {\color  {Chapter }IndeterminatesOfExteriorRing}}{21}{subsection.3.4.19}}
\newlabel{IndeterminatesOfExteriorRing}{{3.4.19}{21}{\textcolor {Chapter }{IndeterminatesOfExteriorRing}}{subsection.3.4.19}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.20}\leavevmode {\color  {Chapter }CoefficientsRing}}{21}{subsection.3.4.20}}
\newlabel{CoefficientsRing}{{3.4.20}{21}{\textcolor {Chapter }{CoefficientsRing}}{subsection.3.4.20}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.21}\leavevmode {\color  {Chapter }KrullDimension}}{21}{subsection.3.4.21}}
\newlabel{KrullDimension}{{3.4.21}{21}{\textcolor {Chapter }{KrullDimension}}{subsection.3.4.21}{}}
\citation{McCRob}
\citation{McCRob}
\citation{McCRob}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.22}\leavevmode {\color  {Chapter }LeftGlobalDimension}}{22}{subsection.3.4.22}}
\newlabel{LeftGlobalDimension}{{3.4.22}{22}{\textcolor {Chapter }{LeftGlobalDimension}}{subsection.3.4.22}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.23}\leavevmode {\color  {Chapter }RightGlobalDimension}}{22}{subsection.3.4.23}}
\newlabel{RightGlobalDimension}{{3.4.23}{22}{\textcolor {Chapter }{RightGlobalDimension}}{subsection.3.4.23}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.24}\leavevmode {\color  {Chapter }GlobalDimension}}{22}{subsection.3.4.24}}
\newlabel{GlobalDimension}{{3.4.24}{22}{\textcolor {Chapter }{GlobalDimension}}{subsection.3.4.24}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.25}\leavevmode {\color  {Chapter }GeneralLinearRank}}{22}{subsection.3.4.25}}
\newlabel{GeneralLinearRank}{{3.4.25}{22}{\textcolor {Chapter }{GeneralLinearRank}}{subsection.3.4.25}{}}
\@writefile{brf}{\backcite{McCRob}{{22}{3.4.25}{subsection.3.4.25}}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.26}\leavevmode {\color  {Chapter }ElementaryRank}}{22}{subsection.3.4.26}}
\newlabel{ElementaryRank}{{3.4.26}{22}{\textcolor {Chapter }{ElementaryRank}}{subsection.3.4.26}{}}
\@writefile{brf}{\backcite{McCRob}{{22}{3.4.26}{subsection.3.4.26}}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.27}\leavevmode {\color  {Chapter }StableRank}}{22}{subsection.3.4.27}}
\newlabel{StableRank}{{3.4.27}{22}{\textcolor {Chapter }{StableRank}}{subsection.3.4.27}{}}
\@writefile{brf}{\backcite{McCRob}{{22}{3.4.27}{subsection.3.4.27}}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.4.28}\leavevmode {\color  {Chapter }AssociatedGradedRing}}{22}{subsection.3.4.28}}
\newlabel{AssociatedGradedRing}{{3.4.28}{22}{\textcolor {Chapter }{AssociatedGradedRing}}{subsection.3.4.28}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.5}\leavevmode {\color  {Chapter }Rings: Operations and Functions}}{22}{section.3.5}}
\newlabel{Rings:Operations}{{3.5}{22}{\textcolor {Chapter }{Rings: Operations and Functions}}{section.3.5}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {4}\leavevmode {\color  {Chapter }Ring Maps}}{23}{chapter.4}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{RingMaps}{{4}{23}{\textcolor {Chapter }{Ring Maps}}{chapter.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.1}\leavevmode {\color  {Chapter }Ring Maps: Category and Representations}}{23}{section.4.1}}
\newlabel{RingMaps:}{{4.1}{23}{\textcolor {Chapter }{Ring Maps: Category and Representations}}{section.4.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1.1}\leavevmode {\color  {Chapter }IsHomalgRingMap}}{23}{subsection.4.1.1}}
\newlabel{IsHomalgRingMap}{{4.1.1}{23}{\textcolor {Chapter }{IsHomalgRingMap}}{subsection.4.1.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1.2}\leavevmode {\color  {Chapter }IsHomalgRingSelfMap}}{23}{subsection.4.1.2}}
\newlabel{IsHomalgRingSelfMap}{{4.1.2}{23}{\textcolor {Chapter }{IsHomalgRingSelfMap}}{subsection.4.1.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1.3}\leavevmode {\color  {Chapter }IsHomalgRingMapRep}}{23}{subsection.4.1.3}}
\newlabel{IsHomalgRingMapRep}{{4.1.3}{23}{\textcolor {Chapter }{IsHomalgRingMapRep}}{subsection.4.1.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.2}\leavevmode {\color  {Chapter }Ring Maps: Constructors}}{23}{section.4.2}}
\newlabel{RingMaps:Constructors}{{4.2}{23}{\textcolor {Chapter }{Ring Maps: Constructors}}{section.4.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color  {Chapter }RingMap (constructor for ring maps)}}{23}{subsection.4.2.1}}
\newlabel{RingMap:constructor for ring maps}{{4.2.1}{23}{\textcolor {Chapter }{RingMap (constructor for ring maps)}}{subsection.4.2.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.3}\leavevmode {\color  {Chapter }Ring Maps: Properties}}{24}{section.4.3}}
\newlabel{RingMaps:Properties}{{4.3}{24}{\textcolor {Chapter }{Ring Maps: Properties}}{section.4.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3.1}\leavevmode {\color  {Chapter }IsMorphism (for ring maps)}}{24}{subsection.4.3.1}}
\newlabel{IsMorphism:for ring maps}{{4.3.1}{24}{\textcolor {Chapter }{IsMorphism (for ring maps)}}{subsection.4.3.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3.2}\leavevmode {\color  {Chapter }IsIdentityMorphism (for ring maps)}}{24}{subsection.4.3.2}}
\newlabel{IsIdentityMorphism:for ring maps}{{4.3.2}{24}{\textcolor {Chapter }{IsIdentityMorphism (for ring maps)}}{subsection.4.3.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3.3}\leavevmode {\color  {Chapter }IsMonomorphism (for ring maps)}}{24}{subsection.4.3.3}}
\newlabel{IsMonomorphism:for ring maps}{{4.3.3}{24}{\textcolor {Chapter }{IsMonomorphism (for ring maps)}}{subsection.4.3.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3.4}\leavevmode {\color  {Chapter }IsEpimorphism (for ring maps)}}{24}{subsection.4.3.4}}
\newlabel{IsEpimorphism:for ring maps}{{4.3.4}{24}{\textcolor {Chapter }{IsEpimorphism (for ring maps)}}{subsection.4.3.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3.5}\leavevmode {\color  {Chapter }IsIsomorphism (for ring maps)}}{24}{subsection.4.3.5}}
\newlabel{IsIsomorphism:for ring maps}{{4.3.5}{24}{\textcolor {Chapter }{IsIsomorphism (for ring maps)}}{subsection.4.3.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3.6}\leavevmode {\color  {Chapter }IsAutomorphism (for ring maps)}}{24}{subsection.4.3.6}}
\newlabel{IsAutomorphism:for ring maps}{{4.3.6}{24}{\textcolor {Chapter }{IsAutomorphism (for ring maps)}}{subsection.4.3.6}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.4}\leavevmode {\color  {Chapter }Ring Maps: Attributes}}{25}{section.4.4}}
\newlabel{RingMaps:Attributes}{{4.4}{25}{\textcolor {Chapter }{Ring Maps: Attributes}}{section.4.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.1}\leavevmode {\color  {Chapter }Source (for ring maps)}}{25}{subsection.4.4.1}}
\newlabel{Source:for ring maps}{{4.4.1}{25}{\textcolor {Chapter }{Source (for ring maps)}}{subsection.4.4.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.2}\leavevmode {\color  {Chapter }Range (for ring maps)}}{25}{subsection.4.4.2}}
\newlabel{Range:for ring maps}{{4.4.2}{25}{\textcolor {Chapter }{Range (for ring maps)}}{subsection.4.4.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.3}\leavevmode {\color  {Chapter }DegreeOfMorphism (for ring maps)}}{25}{subsection.4.4.3}}
\newlabel{DegreeOfMorphism:for ring maps}{{4.4.3}{25}{\textcolor {Chapter }{DegreeOfMorphism (for ring maps)}}{subsection.4.4.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.4}\leavevmode {\color  {Chapter }CoordinateRingOfGraph (for ring maps)}}{25}{subsection.4.4.4}}
\newlabel{CoordinateRingOfGraph:for ring maps}{{4.4.4}{25}{\textcolor {Chapter }{CoordinateRingOfGraph (for ring maps)}}{subsection.4.4.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4.5}\leavevmode {\color  {Chapter }Ring Maps: Operations and Functions}}{25}{section.4.5}}
\newlabel{RingMaps:Operations and Functions}{{4.5}{25}{\textcolor {Chapter }{Ring Maps: Operations and Functions}}{section.4.5}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {5}\leavevmode {\color  {Chapter }Matrices}}{26}{chapter.5}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{Matrices}{{5}{26}{\textcolor {Chapter }{Matrices}}{chapter.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.1}\leavevmode {\color  {Chapter }Matrices: Category and Representations}}{26}{section.5.1}}
\newlabel{Matrices:Category}{{5.1}{26}{\textcolor {Chapter }{Matrices: Category and Representations}}{section.5.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1.1}\leavevmode {\color  {Chapter }IsHomalgMatrix}}{26}{subsection.5.1.1}}
\newlabel{IsHomalgMatrix}{{5.1.1}{26}{\textcolor {Chapter }{IsHomalgMatrix}}{subsection.5.1.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1.2}\leavevmode {\color  {Chapter }IsHomalgInternalMatrixRep}}{26}{subsection.5.1.2}}
\newlabel{IsHomalgInternalMatrixRep}{{5.1.2}{26}{\textcolor {Chapter }{IsHomalgInternalMatrixRep}}{subsection.5.1.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.2}\leavevmode {\color  {Chapter }Matrices: Constructors}}{26}{section.5.2}}
\newlabel{Matrices:Constructors}{{5.2}{26}{\textcolor {Chapter }{Matrices: Constructors}}{section.5.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color  {Chapter }HomalgInitialMatrix (constructor for initial matrices filled with zeros)}}{26}{subsection.5.2.1}}
\newlabel{HomalgInitialMatrix:constructor for initial matrices filled with zeros}{{5.2.1}{26}{\textcolor {Chapter }{HomalgInitialMatrix (constructor for initial matrices filled with zeros)}}{subsection.5.2.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2.2}\leavevmode {\color  {Chapter }HomalgInitialIdentityMatrix (constructor for initial quadratic matrices with ones on the diagonal)}}{27}{subsection.5.2.2}}
\newlabel{HomalgInitialIdentityMatrix:constructor for initial quadratic matrices with ones on the diagonal}{{5.2.2}{27}{\textcolor {Chapter }{HomalgInitialIdentityMatrix (constructor for initial quadratic matrices with ones on the diagonal)}}{subsection.5.2.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2.3}\leavevmode {\color  {Chapter }HomalgZeroMatrix (constructor for zero matrices)}}{28}{subsection.5.2.3}}
\newlabel{HomalgZeroMatrix:constructor for zero matrices}{{5.2.3}{28}{\textcolor {Chapter }{HomalgZeroMatrix (constructor for zero matrices)}}{subsection.5.2.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2.4}\leavevmode {\color  {Chapter }HomalgIdentityMatrix (constructor for identity matrices)}}{28}{subsection.5.2.4}}
\newlabel{HomalgIdentityMatrix:constructor for identity matrices}{{5.2.4}{28}{\textcolor {Chapter }{HomalgIdentityMatrix (constructor for identity matrices)}}{subsection.5.2.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2.5}\leavevmode {\color  {Chapter }HomalgVoidMatrix (constructor for void matrices)}}{28}{subsection.5.2.5}}
\newlabel{HomalgVoidMatrix:constructor for void matrices}{{5.2.5}{28}{\textcolor {Chapter }{HomalgVoidMatrix (constructor for void matrices)}}{subsection.5.2.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2.6}\leavevmode {\color  {Chapter }HomalgMatrix (constructor for matrices using a listlist)}}{29}{subsection.5.2.6}}
\newlabel{HomalgMatrix:constructor for matrices using a listlist}{{5.2.6}{29}{\textcolor {Chapter }{HomalgMatrix (constructor for matrices using a listlist)}}{subsection.5.2.6}{}}
\newlabel{HomalgMatrix:constructor for matrices using a list}{{5.2.6}{29}{\textcolor {Chapter }{HomalgMatrix (constructor for matrices using a listlist)}}{subsection.5.2.6}{}}
\newlabel{HomalgMatrix:constructor for matrices using a string of a listlist}{{5.2.6}{29}{\textcolor {Chapter }{HomalgMatrix (constructor for matrices using a listlist)}}{subsection.5.2.6}{}}
\newlabel{HomalgMatrix:constructor for matrices using a string of a list}{{5.2.6}{29}{\textcolor {Chapter }{HomalgMatrix (constructor for matrices using a listlist)}}{subsection.5.2.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2.7}\leavevmode {\color  {Chapter }HomalgDiagonalMatrix (constructor for diagonal matrices)}}{30}{subsection.5.2.7}}
\newlabel{HomalgDiagonalMatrix:constructor for diagonal matrices}{{5.2.7}{30}{\textcolor {Chapter }{HomalgDiagonalMatrix (constructor for diagonal matrices)}}{subsection.5.2.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2.8}\leavevmode {\color  {Chapter }\texttt  {\char 92\relax }* (copy a matrix over a different ring)}}{30}{subsection.5.2.8}}
\newlabel{*:copy a matrix over a different ring}{{5.2.8}{30}{\textcolor {Chapter }{\texttt {\symbol {92}}* (copy a matrix over a different ring)}}{subsection.5.2.8}{}}
\newlabel{*:copy a matrix over a different ring (right)}{{5.2.8}{30}{\textcolor {Chapter }{\texttt {\symbol {92}}* (copy a matrix over a different ring)}}{subsection.5.2.8}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.3}\leavevmode {\color  {Chapter }Matrices: Properties}}{31}{section.5.3}}
\newlabel{Matrices:Properties}{{5.3}{31}{\textcolor {Chapter }{Matrices: Properties}}{section.5.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.1}\leavevmode {\color  {Chapter }IsZero (for matrices)}}{31}{subsection.5.3.1}}
\newlabel{IsZero:for matrices}{{5.3.1}{31}{\textcolor {Chapter }{IsZero (for matrices)}}{subsection.5.3.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.2}\leavevmode {\color  {Chapter }IsOne}}{31}{subsection.5.3.2}}
\newlabel{IsOne}{{5.3.2}{31}{\textcolor {Chapter }{IsOne}}{subsection.5.3.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.3}\leavevmode {\color  {Chapter }IsUnitFree}}{31}{subsection.5.3.3}}
\newlabel{IsUnitFree}{{5.3.3}{31}{\textcolor {Chapter }{IsUnitFree}}{subsection.5.3.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.4}\leavevmode {\color  {Chapter }IsPermutationMatrix}}{32}{subsection.5.3.4}}
\newlabel{IsPermutationMatrix}{{5.3.4}{32}{\textcolor {Chapter }{IsPermutationMatrix}}{subsection.5.3.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.5}\leavevmode {\color  {Chapter }IsSpecialSubidentityMatrix}}{32}{subsection.5.3.5}}
\newlabel{IsSpecialSubidentityMatrix}{{5.3.5}{32}{\textcolor {Chapter }{IsSpecialSubidentityMatrix}}{subsection.5.3.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.6}\leavevmode {\color  {Chapter }IsSubidentityMatrix}}{32}{subsection.5.3.6}}
\newlabel{IsSubidentityMatrix}{{5.3.6}{32}{\textcolor {Chapter }{IsSubidentityMatrix}}{subsection.5.3.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.7}\leavevmode {\color  {Chapter }IsLeftRegular}}{32}{subsection.5.3.7}}
\newlabel{IsLeftRegular}{{5.3.7}{32}{\textcolor {Chapter }{IsLeftRegular}}{subsection.5.3.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.8}\leavevmode {\color  {Chapter }IsRightRegular}}{32}{subsection.5.3.8}}
\newlabel{IsRightRegular}{{5.3.8}{32}{\textcolor {Chapter }{IsRightRegular}}{subsection.5.3.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.9}\leavevmode {\color  {Chapter }IsInvertibleMatrix}}{32}{subsection.5.3.9}}
\newlabel{IsInvertibleMatrix}{{5.3.9}{32}{\textcolor {Chapter }{IsInvertibleMatrix}}{subsection.5.3.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.10}\leavevmode {\color  {Chapter }IsLeftInvertibleMatrix}}{32}{subsection.5.3.10}}
\newlabel{IsLeftInvertibleMatrix}{{5.3.10}{32}{\textcolor {Chapter }{IsLeftInvertibleMatrix}}{subsection.5.3.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.11}\leavevmode {\color  {Chapter }IsRightInvertibleMatrix}}{32}{subsection.5.3.11}}
\newlabel{IsRightInvertibleMatrix}{{5.3.11}{32}{\textcolor {Chapter }{IsRightInvertibleMatrix}}{subsection.5.3.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.12}\leavevmode {\color  {Chapter }IsEmptyMatrix}}{33}{subsection.5.3.12}}
\newlabel{IsEmptyMatrix}{{5.3.12}{33}{\textcolor {Chapter }{IsEmptyMatrix}}{subsection.5.3.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.13}\leavevmode {\color  {Chapter }IsDiagonalMatrix}}{33}{subsection.5.3.13}}
\newlabel{IsDiagonalMatrix}{{5.3.13}{33}{\textcolor {Chapter }{IsDiagonalMatrix}}{subsection.5.3.13}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.14}\leavevmode {\color  {Chapter }IsScalarlMatrix}}{33}{subsection.5.3.14}}
\newlabel{IsScalarlMatrix}{{5.3.14}{33}{\textcolor {Chapter }{IsScalarlMatrix}}{subsection.5.3.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.15}\leavevmode {\color  {Chapter }IsUpperTriangularMatrix}}{33}{subsection.5.3.15}}
\newlabel{IsUpperTriangularMatrix}{{5.3.15}{33}{\textcolor {Chapter }{IsUpperTriangularMatrix}}{subsection.5.3.15}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.16}\leavevmode {\color  {Chapter }IsLowerTriangularMatrix}}{33}{subsection.5.3.16}}
\newlabel{IsLowerTriangularMatrix}{{5.3.16}{33}{\textcolor {Chapter }{IsLowerTriangularMatrix}}{subsection.5.3.16}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.17}\leavevmode {\color  {Chapter }IsStrictUpperTriangularMatrix}}{33}{subsection.5.3.17}}
\newlabel{IsStrictUpperTriangularMatrix}{{5.3.17}{33}{\textcolor {Chapter }{IsStrictUpperTriangularMatrix}}{subsection.5.3.17}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.18}\leavevmode {\color  {Chapter }IsStrictLowerTriangularMatrix}}{33}{subsection.5.3.18}}
\newlabel{IsStrictLowerTriangularMatrix}{{5.3.18}{33}{\textcolor {Chapter }{IsStrictLowerTriangularMatrix}}{subsection.5.3.18}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.19}\leavevmode {\color  {Chapter }IsUpperStairCaseMatrix}}{33}{subsection.5.3.19}}
\newlabel{IsUpperStairCaseMatrix}{{5.3.19}{33}{\textcolor {Chapter }{IsUpperStairCaseMatrix}}{subsection.5.3.19}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.20}\leavevmode {\color  {Chapter }IsLowerStairCaseMatrix}}{34}{subsection.5.3.20}}
\newlabel{IsLowerStairCaseMatrix}{{5.3.20}{34}{\textcolor {Chapter }{IsLowerStairCaseMatrix}}{subsection.5.3.20}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.21}\leavevmode {\color  {Chapter }IsTriangularMatrix}}{34}{subsection.5.3.21}}
\newlabel{IsTriangularMatrix}{{5.3.21}{34}{\textcolor {Chapter }{IsTriangularMatrix}}{subsection.5.3.21}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.22}\leavevmode {\color  {Chapter }IsBasisOfRowsMatrix}}{34}{subsection.5.3.22}}
\newlabel{IsBasisOfRowsMatrix}{{5.3.22}{34}{\textcolor {Chapter }{IsBasisOfRowsMatrix}}{subsection.5.3.22}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.23}\leavevmode {\color  {Chapter }IsBasisOfColumnsMatrix}}{34}{subsection.5.3.23}}
\newlabel{IsBasisOfColumnsMatrix}{{5.3.23}{34}{\textcolor {Chapter }{IsBasisOfColumnsMatrix}}{subsection.5.3.23}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.24}\leavevmode {\color  {Chapter }IsReducedBasisOfRowsMatrix}}{34}{subsection.5.3.24}}
\newlabel{IsReducedBasisOfRowsMatrix}{{5.3.24}{34}{\textcolor {Chapter }{IsReducedBasisOfRowsMatrix}}{subsection.5.3.24}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.25}\leavevmode {\color  {Chapter }IsReducedBasisOfColumnsMatrix}}{34}{subsection.5.3.25}}
\newlabel{IsReducedBasisOfColumnsMatrix}{{5.3.25}{34}{\textcolor {Chapter }{IsReducedBasisOfColumnsMatrix}}{subsection.5.3.25}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.26}\leavevmode {\color  {Chapter }IsInitialMatrix}}{34}{subsection.5.3.26}}
\newlabel{IsInitialMatrix}{{5.3.26}{34}{\textcolor {Chapter }{IsInitialMatrix}}{subsection.5.3.26}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.27}\leavevmode {\color  {Chapter }IsInitialIdentityMatrix}}{34}{subsection.5.3.27}}
\newlabel{IsInitialIdentityMatrix}{{5.3.27}{34}{\textcolor {Chapter }{IsInitialIdentityMatrix}}{subsection.5.3.27}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3.28}\leavevmode {\color  {Chapter }IsVoidMatrix}}{35}{subsection.5.3.28}}
\newlabel{IsVoidMatrix}{{5.3.28}{35}{\textcolor {Chapter }{IsVoidMatrix}}{subsection.5.3.28}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.4}\leavevmode {\color  {Chapter }Matrices: Attributes}}{35}{section.5.4}}
\newlabel{Matrices:Attributes}{{5.4}{35}{\textcolor {Chapter }{Matrices: Attributes}}{section.5.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.1}\leavevmode {\color  {Chapter }NrRows}}{35}{subsection.5.4.1}}
\newlabel{NrRows}{{5.4.1}{35}{\textcolor {Chapter }{NrRows}}{subsection.5.4.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.2}\leavevmode {\color  {Chapter }NrColumns}}{35}{subsection.5.4.2}}
\newlabel{NrColumns}{{5.4.2}{35}{\textcolor {Chapter }{NrColumns}}{subsection.5.4.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.3}\leavevmode {\color  {Chapter }DeterminantMat}}{35}{subsection.5.4.3}}
\newlabel{DeterminantMat}{{5.4.3}{35}{\textcolor {Chapter }{DeterminantMat}}{subsection.5.4.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.4}\leavevmode {\color  {Chapter }ZeroRows}}{35}{subsection.5.4.4}}
\newlabel{ZeroRows}{{5.4.4}{35}{\textcolor {Chapter }{ZeroRows}}{subsection.5.4.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.5}\leavevmode {\color  {Chapter }ZeroColumns}}{35}{subsection.5.4.5}}
\newlabel{ZeroColumns}{{5.4.5}{35}{\textcolor {Chapter }{ZeroColumns}}{subsection.5.4.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.6}\leavevmode {\color  {Chapter }NonZeroRows}}{35}{subsection.5.4.6}}
\newlabel{NonZeroRows}{{5.4.6}{35}{\textcolor {Chapter }{NonZeroRows}}{subsection.5.4.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.7}\leavevmode {\color  {Chapter }NonZeroColumns}}{36}{subsection.5.4.7}}
\newlabel{NonZeroColumns}{{5.4.7}{36}{\textcolor {Chapter }{NonZeroColumns}}{subsection.5.4.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.8}\leavevmode {\color  {Chapter }PositionOfFirstNonZeroEntryPerRow}}{36}{subsection.5.4.8}}
\newlabel{PositionOfFirstNonZeroEntryPerRow}{{5.4.8}{36}{\textcolor {Chapter }{PositionOfFirstNonZeroEntryPerRow}}{subsection.5.4.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.9}\leavevmode {\color  {Chapter }PositionOfFirstNonZeroEntryPerColumn}}{36}{subsection.5.4.9}}
\newlabel{PositionOfFirstNonZeroEntryPerColumn}{{5.4.9}{36}{\textcolor {Chapter }{PositionOfFirstNonZeroEntryPerColumn}}{subsection.5.4.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.10}\leavevmode {\color  {Chapter }RowRankOfMatrix}}{36}{subsection.5.4.10}}
\newlabel{RowRankOfMatrix}{{5.4.10}{36}{\textcolor {Chapter }{RowRankOfMatrix}}{subsection.5.4.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.11}\leavevmode {\color  {Chapter }ColumnRankOfMatrix}}{36}{subsection.5.4.11}}
\newlabel{ColumnRankOfMatrix}{{5.4.11}{36}{\textcolor {Chapter }{ColumnRankOfMatrix}}{subsection.5.4.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.12}\leavevmode {\color  {Chapter }LeftInverse}}{36}{subsection.5.4.12}}
\newlabel{LeftInverse}{{5.4.12}{36}{\textcolor {Chapter }{LeftInverse}}{subsection.5.4.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.13}\leavevmode {\color  {Chapter }RightInverse}}{36}{subsection.5.4.13}}
\newlabel{RightInverse}{{5.4.13}{36}{\textcolor {Chapter }{RightInverse}}{subsection.5.4.13}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.14}\leavevmode {\color  {Chapter }CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries}}{37}{subsection.5.4.14}}
\newlabel{CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries}{{5.4.14}{37}{\textcolor {Chapter }{CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries}}{subsection.5.4.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.15}\leavevmode {\color  {Chapter }CoefficientsOfNumeratorOfHilbertPoincareSeries}}{37}{subsection.5.4.15}}
\newlabel{CoefficientsOfNumeratorOfHilbertPoincareSeries}{{5.4.15}{37}{\textcolor {Chapter }{CoefficientsOfNumeratorOfHilbertPoincareSeries}}{subsection.5.4.15}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.16}\leavevmode {\color  {Chapter }UnreducedNumeratorOfHilbertPoincareSeries}}{37}{subsection.5.4.16}}
\newlabel{UnreducedNumeratorOfHilbertPoincareSeries}{{5.4.16}{37}{\textcolor {Chapter }{UnreducedNumeratorOfHilbertPoincareSeries}}{subsection.5.4.16}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.17}\leavevmode {\color  {Chapter }NumeratorOfHilbertPoincareSeries}}{37}{subsection.5.4.17}}
\newlabel{NumeratorOfHilbertPoincareSeries}{{5.4.17}{37}{\textcolor {Chapter }{NumeratorOfHilbertPoincareSeries}}{subsection.5.4.17}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.18}\leavevmode {\color  {Chapter }HilbertPoincareSeries}}{37}{subsection.5.4.18}}
\newlabel{HilbertPoincareSeries}{{5.4.18}{37}{\textcolor {Chapter }{HilbertPoincareSeries}}{subsection.5.4.18}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.19}\leavevmode {\color  {Chapter }HilbertPolynomial}}{37}{subsection.5.4.19}}
\newlabel{HilbertPolynomial}{{5.4.19}{37}{\textcolor {Chapter }{HilbertPolynomial}}{subsection.5.4.19}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.20}\leavevmode {\color  {Chapter }AffineDimension}}{37}{subsection.5.4.20}}
\newlabel{AffineDimension}{{5.4.20}{37}{\textcolor {Chapter }{AffineDimension}}{subsection.5.4.20}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.21}\leavevmode {\color  {Chapter }AffineDegree}}{37}{subsection.5.4.21}}
\newlabel{AffineDegree}{{5.4.21}{37}{\textcolor {Chapter }{AffineDegree}}{subsection.5.4.21}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.22}\leavevmode {\color  {Chapter }ProjectiveDegree}}{38}{subsection.5.4.22}}
\newlabel{ProjectiveDegree}{{5.4.22}{38}{\textcolor {Chapter }{ProjectiveDegree}}{subsection.5.4.22}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.23}\leavevmode {\color  {Chapter }ConstantTermOfHilbertPolynomialn}}{38}{subsection.5.4.23}}
\newlabel{ConstantTermOfHilbertPolynomialn}{{5.4.23}{38}{\textcolor {Chapter }{ConstantTermOfHilbertPolynomialn}}{subsection.5.4.23}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4.24}\leavevmode {\color  {Chapter }MatrixOfSymbols}}{38}{subsection.5.4.24}}
\newlabel{MatrixOfSymbols}{{5.4.24}{38}{\textcolor {Chapter }{MatrixOfSymbols}}{subsection.5.4.24}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5.5}\leavevmode {\color  {Chapter }Matrices: Operations and Functions}}{38}{section.5.5}}
\newlabel{Matrices:Operations}{{5.5}{38}{\textcolor {Chapter }{Matrices: Operations and Functions}}{section.5.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.1}\leavevmode {\color  {Chapter }HomalgRing (for matrices)}}{38}{subsection.5.5.1}}
\newlabel{HomalgRing:for matrices}{{5.5.1}{38}{\textcolor {Chapter }{HomalgRing (for matrices)}}{subsection.5.5.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.2}\leavevmode {\color  {Chapter }LeftInverse (for matrices)}}{38}{subsection.5.5.2}}
\newlabel{LeftInverse:for matrices}{{5.5.2}{38}{\textcolor {Chapter }{LeftInverse (for matrices)}}{subsection.5.5.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.3}\leavevmode {\color  {Chapter }RightInverse (for matrices)}}{39}{subsection.5.5.3}}
\newlabel{RightInverse:for matrices}{{5.5.3}{39}{\textcolor {Chapter }{RightInverse (for matrices)}}{subsection.5.5.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.4}\leavevmode {\color  {Chapter }LeftInverseLazy (for matrices)}}{40}{subsection.5.5.4}}
\newlabel{LeftInverseLazy:for matrices}{{5.5.4}{40}{\textcolor {Chapter }{LeftInverseLazy (for matrices)}}{subsection.5.5.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.5}\leavevmode {\color  {Chapter }RightInverseLazy (for matrices)}}{40}{subsection.5.5.5}}
\newlabel{RightInverseLazy:for matrices}{{5.5.5}{40}{\textcolor {Chapter }{RightInverseLazy (for matrices)}}{subsection.5.5.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.6}\leavevmode {\color  {Chapter }Involution (for matrices)}}{40}{subsection.5.5.6}}
\newlabel{Involution:for matrices}{{5.5.6}{40}{\textcolor {Chapter }{Involution (for matrices)}}{subsection.5.5.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.7}\leavevmode {\color  {Chapter }CertainRows (for matrices)}}{41}{subsection.5.5.7}}
\newlabel{CertainRows:for matrices}{{5.5.7}{41}{\textcolor {Chapter }{CertainRows (for matrices)}}{subsection.5.5.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.8}\leavevmode {\color  {Chapter }CertainColumns (for matrices)}}{41}{subsection.5.5.8}}
\newlabel{CertainColumns:for matrices}{{5.5.8}{41}{\textcolor {Chapter }{CertainColumns (for matrices)}}{subsection.5.5.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.9}\leavevmode {\color  {Chapter }UnionOfRows (for matrices)}}{41}{subsection.5.5.9}}
\newlabel{UnionOfRows:for matrices}{{5.5.9}{41}{\textcolor {Chapter }{UnionOfRows (for matrices)}}{subsection.5.5.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.10}\leavevmode {\color  {Chapter }UnionOfColumns (for matrices)}}{41}{subsection.5.5.10}}
\newlabel{UnionOfColumns:for matrices}{{5.5.10}{41}{\textcolor {Chapter }{UnionOfColumns (for matrices)}}{subsection.5.5.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.11}\leavevmode {\color  {Chapter }DiagMat (for matrices)}}{41}{subsection.5.5.11}}
\newlabel{DiagMat:for matrices}{{5.5.11}{41}{\textcolor {Chapter }{DiagMat (for matrices)}}{subsection.5.5.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.12}\leavevmode {\color  {Chapter }KroneckerMat (for matrices)}}{41}{subsection.5.5.12}}
\newlabel{KroneckerMat:for matrices}{{5.5.12}{41}{\textcolor {Chapter }{KroneckerMat (for matrices)}}{subsection.5.5.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.13}\leavevmode {\color  {Chapter }\texttt  {\char 92\relax }* (for ring elements and matrices)}}{42}{subsection.5.5.13}}
\newlabel{*:for ring elements and matrices}{{5.5.13}{42}{\textcolor {Chapter }{\texttt {\symbol {92}}* (for ring elements and matrices)}}{subsection.5.5.13}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.14}\leavevmode {\color  {Chapter }\texttt  {\char 92\relax }+ (for matrices)}}{42}{subsection.5.5.14}}
\newlabel{+:for matrices}{{5.5.14}{42}{\textcolor {Chapter }{\texttt {\symbol {92}}+ (for matrices)}}{subsection.5.5.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.15}\leavevmode {\color  {Chapter }\texttt  {\char 92\relax }- (for matrices)}}{42}{subsection.5.5.15}}
\newlabel{-:for matrices}{{5.5.15}{42}{\textcolor {Chapter }{\texttt {\symbol {92}}- (for matrices)}}{subsection.5.5.15}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.16}\leavevmode {\color  {Chapter }\texttt  {\char 92\relax }* (for composable matrices)}}{42}{subsection.5.5.16}}
\newlabel{*:for composable matrices}{{5.5.16}{42}{\textcolor {Chapter }{\texttt {\symbol {92}}* (for composable matrices)}}{subsection.5.5.16}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.17}\leavevmode {\color  {Chapter }\texttt  {\char 92\relax }= (for matrices)}}{42}{subsection.5.5.17}}
\newlabel{=:for matrices}{{5.5.17}{42}{\textcolor {Chapter }{\texttt {\symbol {92}}= (for matrices)}}{subsection.5.5.17}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.18}\leavevmode {\color  {Chapter }GetColumnIndependentUnitPositions (for matrices)}}{43}{subsection.5.5.18}}
\newlabel{GetColumnIndependentUnitPositions:for matrices}{{5.5.18}{43}{\textcolor {Chapter }{GetColumnIndependentUnitPositions (for matrices)}}{subsection.5.5.18}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.19}\leavevmode {\color  {Chapter }GetRowIndependentUnitPositions (for matrices)}}{43}{subsection.5.5.19}}
\newlabel{GetRowIndependentUnitPositions:for matrices}{{5.5.19}{43}{\textcolor {Chapter }{GetRowIndependentUnitPositions (for matrices)}}{subsection.5.5.19}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.20}\leavevmode {\color  {Chapter }GetUnitPosition (for matrices)}}{43}{subsection.5.5.20}}
\newlabel{GetUnitPosition:for matrices}{{5.5.20}{43}{\textcolor {Chapter }{GetUnitPosition (for matrices)}}{subsection.5.5.20}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.21}\leavevmode {\color  {Chapter }Eliminate}}{43}{subsection.5.5.21}}
\newlabel{Eliminate}{{5.5.21}{43}{\textcolor {Chapter }{Eliminate}}{subsection.5.5.21}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.22}\leavevmode {\color  {Chapter }BasisOfRowModule (for matrices)}}{44}{subsection.5.5.22}}
\newlabel{BasisOfRowModule:for matrices}{{5.5.22}{44}{\textcolor {Chapter }{BasisOfRowModule (for matrices)}}{subsection.5.5.22}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.23}\leavevmode {\color  {Chapter }BasisOfColumnModule (for matrices)}}{44}{subsection.5.5.23}}
\newlabel{BasisOfColumnModule:for matrices}{{5.5.23}{44}{\textcolor {Chapter }{BasisOfColumnModule (for matrices)}}{subsection.5.5.23}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.24}\leavevmode {\color  {Chapter }DecideZeroRows (for pairs of matrices)}}{44}{subsection.5.5.24}}
\newlabel{DecideZeroRows:for pairs of matrices}{{5.5.24}{44}{\textcolor {Chapter }{DecideZeroRows (for pairs of matrices)}}{subsection.5.5.24}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.25}\leavevmode {\color  {Chapter }DecideZeroColumns (for pairs of matrices)}}{44}{subsection.5.5.25}}
\newlabel{DecideZeroColumns:for pairs of matrices}{{5.5.25}{44}{\textcolor {Chapter }{DecideZeroColumns (for pairs of matrices)}}{subsection.5.5.25}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.26}\leavevmode {\color  {Chapter }SyzygiesGeneratorsOfRows (for matrices)}}{45}{subsection.5.5.26}}
\newlabel{SyzygiesGeneratorsOfRows:for matrices}{{5.5.26}{45}{\textcolor {Chapter }{SyzygiesGeneratorsOfRows (for matrices)}}{subsection.5.5.26}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.27}\leavevmode {\color  {Chapter }SyzygiesGeneratorsOfColumns (for matrices)}}{45}{subsection.5.5.27}}
\newlabel{SyzygiesGeneratorsOfColumns:for matrices}{{5.5.27}{45}{\textcolor {Chapter }{SyzygiesGeneratorsOfColumns (for matrices)}}{subsection.5.5.27}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.28}\leavevmode {\color  {Chapter }SyzygiesGeneratorsOfRows (for pairs of matrices)}}{45}{subsection.5.5.28}}
\newlabel{SyzygiesGeneratorsOfRows:for pairs of matrices}{{5.5.28}{45}{\textcolor {Chapter }{SyzygiesGeneratorsOfRows (for pairs of matrices)}}{subsection.5.5.28}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.29}\leavevmode {\color  {Chapter }SyzygiesGeneratorsOfColumns (for pairs of matrices)}}{45}{subsection.5.5.29}}
\newlabel{SyzygiesGeneratorsOfColumns:for pairs of matrices}{{5.5.29}{45}{\textcolor {Chapter }{SyzygiesGeneratorsOfColumns (for pairs of matrices)}}{subsection.5.5.29}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.30}\leavevmode {\color  {Chapter }ReducedBasisOfRowModule (for matrices)}}{45}{subsection.5.5.30}}
\newlabel{ReducedBasisOfRowModule:for matrices}{{5.5.30}{45}{\textcolor {Chapter }{ReducedBasisOfRowModule (for matrices)}}{subsection.5.5.30}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.31}\leavevmode {\color  {Chapter }ReducedBasisOfColumnModule (for matrices)}}{45}{subsection.5.5.31}}
\newlabel{ReducedBasisOfColumnModule:for matrices}{{5.5.31}{45}{\textcolor {Chapter }{ReducedBasisOfColumnModule (for matrices)}}{subsection.5.5.31}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.32}\leavevmode {\color  {Chapter }ReducedSyzygiesGeneratorsOfRows (for matrices)}}{46}{subsection.5.5.32}}
\newlabel{ReducedSyzygiesGeneratorsOfRows:for matrices}{{5.5.32}{46}{\textcolor {Chapter }{ReducedSyzygiesGeneratorsOfRows (for matrices)}}{subsection.5.5.32}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.33}\leavevmode {\color  {Chapter }ReducedSyzygiesGeneratorsOfColumns (for matrices)}}{46}{subsection.5.5.33}}
\newlabel{ReducedSyzygiesGeneratorsOfColumns:for matrices}{{5.5.33}{46}{\textcolor {Chapter }{ReducedSyzygiesGeneratorsOfColumns (for matrices)}}{subsection.5.5.33}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.34}\leavevmode {\color  {Chapter }BasisOfRowsCoeff (for matrices)}}{46}{subsection.5.5.34}}
\newlabel{BasisOfRowsCoeff:for matrices}{{5.5.34}{46}{\textcolor {Chapter }{BasisOfRowsCoeff (for matrices)}}{subsection.5.5.34}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.35}\leavevmode {\color  {Chapter }BasisOfColumnsCoeff (for matrices)}}{46}{subsection.5.5.35}}
\newlabel{BasisOfColumnsCoeff:for matrices}{{5.5.35}{46}{\textcolor {Chapter }{BasisOfColumnsCoeff (for matrices)}}{subsection.5.5.35}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.36}\leavevmode {\color  {Chapter }DecideZeroRowsEffectively (for pairs of matrices)}}{46}{subsection.5.5.36}}
\newlabel{DecideZeroRowsEffectively:for pairs of matrices}{{5.5.36}{46}{\textcolor {Chapter }{DecideZeroRowsEffectively (for pairs of matrices)}}{subsection.5.5.36}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.37}\leavevmode {\color  {Chapter }DecideZeroColumnsEffectively (for pairs of matrices)}}{47}{subsection.5.5.37}}
\newlabel{DecideZeroColumnsEffectively:for pairs of matrices}{{5.5.37}{47}{\textcolor {Chapter }{DecideZeroColumnsEffectively (for pairs of matrices)}}{subsection.5.5.37}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.38}\leavevmode {\color  {Chapter }BasisOfRows (for matrices)}}{47}{subsection.5.5.38}}
\newlabel{BasisOfRows:for matrices}{{5.5.38}{47}{\textcolor {Chapter }{BasisOfRows (for matrices)}}{subsection.5.5.38}{}}
\newlabel{BasisOfRows:for pairs of matrices}{{5.5.38}{47}{\textcolor {Chapter }{BasisOfRows (for matrices)}}{subsection.5.5.38}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.39}\leavevmode {\color  {Chapter }BasisOfColumns (for matrices)}}{47}{subsection.5.5.39}}
\newlabel{BasisOfColumns:for matrices}{{5.5.39}{47}{\textcolor {Chapter }{BasisOfColumns (for matrices)}}{subsection.5.5.39}{}}
\newlabel{BasisOfColumns:for pairs of matrices}{{5.5.39}{47}{\textcolor {Chapter }{BasisOfColumns (for matrices)}}{subsection.5.5.39}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.40}\leavevmode {\color  {Chapter }DecideZero (for matrices and relations)}}{47}{subsection.5.5.40}}
\newlabel{DecideZero:for matrices and relations}{{5.5.40}{47}{\textcolor {Chapter }{DecideZero (for matrices and relations)}}{subsection.5.5.40}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.41}\leavevmode {\color  {Chapter }SyzygiesOfRows (for matrices)}}{47}{subsection.5.5.41}}
\newlabel{SyzygiesOfRows:for matrices}{{5.5.41}{47}{\textcolor {Chapter }{SyzygiesOfRows (for matrices)}}{subsection.5.5.41}{}}
\newlabel{SyzygiesOfRows:for pairs of matrices}{{5.5.41}{47}{\textcolor {Chapter }{SyzygiesOfRows (for matrices)}}{subsection.5.5.41}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.42}\leavevmode {\color  {Chapter }SyzygiesOfColumns (for matrices)}}{48}{subsection.5.5.42}}
\newlabel{SyzygiesOfColumns:for matrices}{{5.5.42}{48}{\textcolor {Chapter }{SyzygiesOfColumns (for matrices)}}{subsection.5.5.42}{}}
\newlabel{SyzygiesOfColumns:for pairs of matrices}{{5.5.42}{48}{\textcolor {Chapter }{SyzygiesOfColumns (for matrices)}}{subsection.5.5.42}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.43}\leavevmode {\color  {Chapter }ReducedSyzygiesOfRows (for matrices)}}{48}{subsection.5.5.43}}
\newlabel{ReducedSyzygiesOfRows:for matrices}{{5.5.43}{48}{\textcolor {Chapter }{ReducedSyzygiesOfRows (for matrices)}}{subsection.5.5.43}{}}
\newlabel{ReducedSyzygiesOfRows:for pairs of matrices}{{5.5.43}{48}{\textcolor {Chapter }{ReducedSyzygiesOfRows (for matrices)}}{subsection.5.5.43}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.44}\leavevmode {\color  {Chapter }ReducedSyzygiesOfColumns (for matrices)}}{48}{subsection.5.5.44}}
\newlabel{ReducedSyzygiesOfColumns:for matrices}{{5.5.44}{48}{\textcolor {Chapter }{ReducedSyzygiesOfColumns (for matrices)}}{subsection.5.5.44}{}}
\newlabel{ReducedSyzygiesOfColumns:for pairs of matrices}{{5.5.44}{48}{\textcolor {Chapter }{ReducedSyzygiesOfColumns (for matrices)}}{subsection.5.5.44}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.45}\leavevmode {\color  {Chapter }RightDivide (for pairs of matrices)}}{48}{subsection.5.5.45}}
\newlabel{RightDivide:for pairs of matrices}{{5.5.45}{48}{\textcolor {Chapter }{RightDivide (for pairs of matrices)}}{subsection.5.5.45}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.46}\leavevmode {\color  {Chapter }LeftDivide (for pairs of matrices)}}{48}{subsection.5.5.46}}
\newlabel{LeftDivide:for pairs of matrices}{{5.5.46}{48}{\textcolor {Chapter }{LeftDivide (for pairs of matrices)}}{subsection.5.5.46}{}}
\citation{BR}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.47}\leavevmode {\color  {Chapter }RightDivide (for triples of matrices)}}{49}{subsection.5.5.47}}
\newlabel{RightDivide:for triples of matrices}{{5.5.47}{49}{\textcolor {Chapter }{RightDivide (for triples of matrices)}}{subsection.5.5.47}{}}
\@writefile{brf}{\backcite{BR}{{49}{5.5.47}{subsection.5.5.47}}}
\citation{BR}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.48}\leavevmode {\color  {Chapter }LeftDivide (for triples of matrices)}}{50}{subsection.5.5.48}}
\newlabel{LeftDivide:for triples of matrices}{{5.5.48}{50}{\textcolor {Chapter }{LeftDivide (for triples of matrices)}}{subsection.5.5.48}{}}
\@writefile{brf}{\backcite{BR}{{50}{5.5.48}{subsection.5.5.48}}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.49}\leavevmode {\color  {Chapter }GenerateSameRowModule (for pairs of matrices)}}{51}{subsection.5.5.49}}
\newlabel{GenerateSameRowModule:for pairs of matrices}{{5.5.49}{51}{\textcolor {Chapter }{GenerateSameRowModule (for pairs of matrices)}}{subsection.5.5.49}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5.50}\leavevmode {\color  {Chapter }GenerateSameColumnModule (for pairs of matrices)}}{51}{subsection.5.5.50}}
\newlabel{GenerateSameColumnModule:for pairs of matrices}{{5.5.50}{51}{\textcolor {Chapter }{GenerateSameColumnModule (for pairs of matrices)}}{subsection.5.5.50}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {6}\leavevmode {\color  {Chapter }Ring Relations}}{52}{chapter.6}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{RingRelations}{{6}{52}{\textcolor {Chapter }{Ring Relations}}{chapter.6}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.1}\leavevmode {\color  {Chapter }Ring Relations: Categories and Representations}}{52}{section.6.1}}
\newlabel{RingRelations:Category}{{6.1}{52}{\textcolor {Chapter }{Ring Relations: Categories and Representations}}{section.6.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1.1}\leavevmode {\color  {Chapter }IsHomalgRingRelations}}{52}{subsection.6.1.1}}
\newlabel{IsHomalgRingRelations}{{6.1.1}{52}{\textcolor {Chapter }{IsHomalgRingRelations}}{subsection.6.1.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1.2}\leavevmode {\color  {Chapter }IsHomalgRingRelationsAsGeneratorsOfLeftIdeal}}{52}{subsection.6.1.2}}
\newlabel{IsHomalgRingRelationsAsGeneratorsOfLeftIdeal}{{6.1.2}{52}{\textcolor {Chapter }{IsHomalgRingRelationsAsGeneratorsOfLeftIdeal}}{subsection.6.1.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1.3}\leavevmode {\color  {Chapter }IsHomalgRingRelationsAsGeneratorsOfRightIdeal}}{52}{subsection.6.1.3}}
\newlabel{IsHomalgRingRelationsAsGeneratorsOfRightIdeal}{{6.1.3}{52}{\textcolor {Chapter }{IsHomalgRingRelationsAsGeneratorsOfRightIdeal}}{subsection.6.1.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1.4}\leavevmode {\color  {Chapter }IsRingRelationsRep}}{52}{subsection.6.1.4}}
\newlabel{IsRingRelationsRep}{{6.1.4}{52}{\textcolor {Chapter }{IsRingRelationsRep}}{subsection.6.1.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.2}\leavevmode {\color  {Chapter }Ring Relations: Constructors}}{53}{section.6.2}}
\newlabel{RingRelations:Constructors}{{6.2}{53}{\textcolor {Chapter }{Ring Relations: Constructors}}{section.6.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.3}\leavevmode {\color  {Chapter }Ring Relations: Properties}}{53}{section.6.3}}
\newlabel{RingRelations:Properties}{{6.3}{53}{\textcolor {Chapter }{Ring Relations: Properties}}{section.6.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.3.1}\leavevmode {\color  {Chapter }CanBeUsedToDecideZero}}{53}{subsection.6.3.1}}
\newlabel{CanBeUsedToDecideZero}{{6.3.1}{53}{\textcolor {Chapter }{CanBeUsedToDecideZero}}{subsection.6.3.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.3.2}\leavevmode {\color  {Chapter }IsInjectivePresentation}}{53}{subsection.6.3.2}}
\newlabel{IsInjectivePresentation}{{6.3.2}{53}{\textcolor {Chapter }{IsInjectivePresentation}}{subsection.6.3.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.4}\leavevmode {\color  {Chapter }Ring Relations: Attributes}}{53}{section.6.4}}
\newlabel{RingRelations:Attributes}{{6.4}{53}{\textcolor {Chapter }{Ring Relations: Attributes}}{section.6.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6.5}\leavevmode {\color  {Chapter }Ring Relations: Operations and Functions}}{53}{section.6.5}}
\newlabel{RingRelations:Operations}{{6.5}{53}{\textcolor {Chapter }{Ring Relations: Operations and Functions}}{section.6.5}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {A}\leavevmode {\color  {Chapter }The Basic Matrix Operations}}{54}{appendix.A}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{Basic_Operations}{{A}{54}{\textcolor {Chapter }{The Basic Matrix Operations}}{appendix.A}{}}
\@writefile{toc}{\contentsline {section}{\numberline {A.1}\leavevmode {\color  {Chapter }Main}}{54}{section.A.1}}
\newlabel{Main}{{A.1}{54}{\textcolor {Chapter }{Main}}{section.A.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {A.2}\leavevmode {\color  {Chapter }Effective}}{54}{section.A.2}}
\newlabel{Effective}{{A.2}{54}{\textcolor {Chapter }{Effective}}{section.A.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {A.3}\leavevmode {\color  {Chapter }Relative}}{54}{section.A.3}}
\newlabel{Relative}{{A.3}{54}{\textcolor {Chapter }{Relative}}{section.A.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {A.4}\leavevmode {\color  {Chapter }Reduced}}{55}{section.A.4}}
\newlabel{Reduced}{{A.4}{55}{\textcolor {Chapter }{Reduced}}{section.A.4}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {B}\leavevmode {\color  {Chapter }The Matrix Tool Operations}}{56}{appendix.B}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{Tool_Operations}{{B}{56}{\textcolor {Chapter }{The Matrix Tool Operations}}{appendix.B}{}}
\@writefile{toc}{\contentsline {section}{\numberline {B.1}\leavevmode {\color  {Chapter }The Tool Operations \emph  {without} a Fallback Method}}{56}{section.B.1}}
\newlabel{ToolsNoFallBack}{{B.1}{56}{\textcolor {Chapter }{The Tool Operations \emph {without} a Fallback Method}}{section.B.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.1}\leavevmode {\color  {Chapter }InitialMatrix (homalgTable entry for initial matrices)}}{56}{subsection.B.1.1}}
\newlabel{InitialMatrix:homalgTable entry for initial matrices}{{B.1.1}{56}{\textcolor {Chapter }{InitialMatrix (homalgTable entry for initial matrices)}}{subsection.B.1.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.2}\leavevmode {\color  {Chapter }InitialIdentityMatrix (homalgTable entry for initial identity matrices)}}{56}{subsection.B.1.2}}
\newlabel{InitialIdentityMatrix:homalgTable entry for initial identity matrices}{{B.1.2}{56}{\textcolor {Chapter }{InitialIdentityMatrix (homalgTable entry for initial identity matrices)}}{subsection.B.1.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.3}\leavevmode {\color  {Chapter }ZeroMatrix (homalgTable entry)}}{56}{subsection.B.1.3}}
\newlabel{ZeroMatrix:homalgTable entry}{{B.1.3}{56}{\textcolor {Chapter }{ZeroMatrix (homalgTable entry)}}{subsection.B.1.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.4}\leavevmode {\color  {Chapter }IdentityMatrix (homalgTable entry)}}{57}{subsection.B.1.4}}
\newlabel{IdentityMatrix:homalgTable entry}{{B.1.4}{57}{\textcolor {Chapter }{IdentityMatrix (homalgTable entry)}}{subsection.B.1.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.5}\leavevmode {\color  {Chapter }Involution (homalgTable entry)}}{57}{subsection.B.1.5}}
\newlabel{Involution:homalgTable entry}{{B.1.5}{57}{\textcolor {Chapter }{Involution (homalgTable entry)}}{subsection.B.1.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.6}\leavevmode {\color  {Chapter }CertainRows (homalgTable entry)}}{57}{subsection.B.1.6}}
\newlabel{CertainRows:homalgTable entry}{{B.1.6}{57}{\textcolor {Chapter }{CertainRows (homalgTable entry)}}{subsection.B.1.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.7}\leavevmode {\color  {Chapter }CertainColumns (homalgTable entry)}}{57}{subsection.B.1.7}}
\newlabel{CertainColumns:homalgTable entry}{{B.1.7}{57}{\textcolor {Chapter }{CertainColumns (homalgTable entry)}}{subsection.B.1.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.8}\leavevmode {\color  {Chapter }UnionOfRows (homalgTable entry)}}{57}{subsection.B.1.8}}
\newlabel{UnionOfRows:homalgTable entry}{{B.1.8}{57}{\textcolor {Chapter }{UnionOfRows (homalgTable entry)}}{subsection.B.1.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.9}\leavevmode {\color  {Chapter }UnionOfColumns (homalgTable entry)}}{57}{subsection.B.1.9}}
\newlabel{UnionOfColumns:homalgTable entry}{{B.1.9}{57}{\textcolor {Chapter }{UnionOfColumns (homalgTable entry)}}{subsection.B.1.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.10}\leavevmode {\color  {Chapter }DiagMat (homalgTable entry)}}{58}{subsection.B.1.10}}
\newlabel{DiagMat:homalgTable entry}{{B.1.10}{58}{\textcolor {Chapter }{DiagMat (homalgTable entry)}}{subsection.B.1.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.11}\leavevmode {\color  {Chapter }KroneckerMat (homalgTable entry)}}{58}{subsection.B.1.11}}
\newlabel{KroneckerMat:homalgTable entry}{{B.1.11}{58}{\textcolor {Chapter }{KroneckerMat (homalgTable entry)}}{subsection.B.1.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.12}\leavevmode {\color  {Chapter }MulMat (homalgTable entry)}}{58}{subsection.B.1.12}}
\newlabel{MulMat:homalgTable entry}{{B.1.12}{58}{\textcolor {Chapter }{MulMat (homalgTable entry)}}{subsection.B.1.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.13}\leavevmode {\color  {Chapter }AddMat (homalgTable entry)}}{58}{subsection.B.1.13}}
\newlabel{AddMat:homalgTable entry}{{B.1.13}{58}{\textcolor {Chapter }{AddMat (homalgTable entry)}}{subsection.B.1.13}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.14}\leavevmode {\color  {Chapter }SubMat (homalgTable entry)}}{58}{subsection.B.1.14}}
\newlabel{SubMat:homalgTable entry}{{B.1.14}{58}{\textcolor {Chapter }{SubMat (homalgTable entry)}}{subsection.B.1.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.15}\leavevmode {\color  {Chapter }Compose (homalgTable entry)}}{58}{subsection.B.1.15}}
\newlabel{Compose:homalgTable entry}{{B.1.15}{58}{\textcolor {Chapter }{Compose (homalgTable entry)}}{subsection.B.1.15}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.16}\leavevmode {\color  {Chapter }IsZeroMatrix (homalgTable entry)}}{59}{subsection.B.1.16}}
\newlabel{IsZeroMatrix:homalgTable entry}{{B.1.16}{59}{\textcolor {Chapter }{IsZeroMatrix (homalgTable entry)}}{subsection.B.1.16}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.17}\leavevmode {\color  {Chapter }NrRows (homalgTable entry)}}{59}{subsection.B.1.17}}
\newlabel{NrRows:homalgTable entry}{{B.1.17}{59}{\textcolor {Chapter }{NrRows (homalgTable entry)}}{subsection.B.1.17}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.18}\leavevmode {\color  {Chapter }NrColumns (homalgTable entry)}}{60}{subsection.B.1.18}}
\newlabel{NrColumns:homalgTable entry}{{B.1.18}{60}{\textcolor {Chapter }{NrColumns (homalgTable entry)}}{subsection.B.1.18}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.1.19}\leavevmode {\color  {Chapter }Determinant (homalgTable entry)}}{61}{subsection.B.1.19}}
\newlabel{Determinant:homalgTable entry}{{B.1.19}{61}{\textcolor {Chapter }{Determinant (homalgTable entry)}}{subsection.B.1.19}{}}
\@writefile{toc}{\contentsline {section}{\numberline {B.2}\leavevmode {\color  {Chapter }The Tool Operations with a Fallback Method}}{62}{section.B.2}}
\newlabel{ToolsFallBack}{{B.2}{62}{\textcolor {Chapter }{The Tool Operations with a Fallback Method}}{section.B.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.1}\leavevmode {\color  {Chapter }AreEqualMatrices (homalgTable entry)}}{62}{subsection.B.2.1}}
\newlabel{AreEqualMatrices:homalgTable entry}{{B.2.1}{62}{\textcolor {Chapter }{AreEqualMatrices (homalgTable entry)}}{subsection.B.2.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.2}\leavevmode {\color  {Chapter }IsIdentityMatrix (homalgTable entry)}}{63}{subsection.B.2.2}}
\newlabel{IsIdentityMatrix:homalgTable entry}{{B.2.2}{63}{\textcolor {Chapter }{IsIdentityMatrix (homalgTable entry)}}{subsection.B.2.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.3}\leavevmode {\color  {Chapter }IsDiagonalMatrix (homalgTable entry)}}{64}{subsection.B.2.3}}
\newlabel{IsDiagonalMatrix:homalgTable entry}{{B.2.3}{64}{\textcolor {Chapter }{IsDiagonalMatrix (homalgTable entry)}}{subsection.B.2.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.4}\leavevmode {\color  {Chapter }ZeroRows (homalgTable entry)}}{65}{subsection.B.2.4}}
\newlabel{ZeroRows:homalgTable entry}{{B.2.4}{65}{\textcolor {Chapter }{ZeroRows (homalgTable entry)}}{subsection.B.2.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.5}\leavevmode {\color  {Chapter }ZeroColumns (homalgTable entry)}}{65}{subsection.B.2.5}}
\newlabel{ZeroColumns:homalgTable entry}{{B.2.5}{65}{\textcolor {Chapter }{ZeroColumns (homalgTable entry)}}{subsection.B.2.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.6}\leavevmode {\color  {Chapter }GetColumnIndependentUnitPositions (homalgTable entry)}}{66}{subsection.B.2.6}}
\newlabel{GetColumnIndependentUnitPositions:homalgTable entry}{{B.2.6}{66}{\textcolor {Chapter }{GetColumnIndependentUnitPositions (homalgTable entry)}}{subsection.B.2.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.7}\leavevmode {\color  {Chapter }GetRowIndependentUnitPositions (homalgTable entry)}}{67}{subsection.B.2.7}}
\newlabel{GetRowIndependentUnitPositions:homalgTable entry}{{B.2.7}{67}{\textcolor {Chapter }{GetRowIndependentUnitPositions (homalgTable entry)}}{subsection.B.2.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.8}\leavevmode {\color  {Chapter }GetUnitPosition (homalgTable entry)}}{68}{subsection.B.2.8}}
\newlabel{GetUnitPosition:homalgTable entry}{{B.2.8}{68}{\textcolor {Chapter }{GetUnitPosition (homalgTable entry)}}{subsection.B.2.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.9}\leavevmode {\color  {Chapter }PositionOfFirstNonZeroEntryPerRow (homalgTable entry)}}{69}{subsection.B.2.9}}
\newlabel{PositionOfFirstNonZeroEntryPerRow:homalgTable entry}{{B.2.9}{69}{\textcolor {Chapter }{PositionOfFirstNonZeroEntryPerRow (homalgTable entry)}}{subsection.B.2.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {B.2.10}\leavevmode {\color  {Chapter }PositionOfFirstNonZeroEntryPerColumn (homalgTable entry)}}{70}{subsection.B.2.10}}
\newlabel{PositionOfFirstNonZeroEntryPerColumn:homalgTable entry}{{B.2.10}{70}{\textcolor {Chapter }{PositionOfFirstNonZeroEntryPerColumn (homalgTable entry)}}{subsection.B.2.10}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {C}\leavevmode {\color  {Chapter }Logic Subpackages}}{72}{appendix.C}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{Logic}{{C}{72}{\textcolor {Chapter }{Logic Subpackages}}{appendix.C}{}}
\@writefile{toc}{\contentsline {section}{\numberline {C.1}\leavevmode {\color  {Chapter }\textsf  {LIRNG}: Logical Implications for Rings}}{72}{section.C.1}}
\newlabel{Rings:LIRNG}{{C.1}{72}{\textcolor {Chapter }{\textsf {LIRNG}: Logical Implications for Rings}}{section.C.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {C.2}\leavevmode {\color  {Chapter }\textsf  {LIMAP}: Logical Implications for Ring Maps}}{72}{section.C.2}}
\newlabel{RingMaps:LIMAP}{{C.2}{72}{\textcolor {Chapter }{\textsf {LIMAP}: Logical Implications for Ring Maps}}{section.C.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {C.3}\leavevmode {\color  {Chapter }\textsf  {LIMAT}: Logical Implications for Matrices}}{72}{section.C.3}}
\newlabel{Matrices:LIMAT}{{C.3}{72}{\textcolor {Chapter }{\textsf {LIMAT}: Logical Implications for Matrices}}{section.C.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {C.4}\leavevmode {\color  {Chapter }\textsf  {COLEM}: Clever Operations for Lazy Evaluated Matrices}}{72}{section.C.4}}
\newlabel{Matrices:COLEM}{{C.4}{72}{\textcolor {Chapter }{\textsf {COLEM}: Clever Operations for Lazy Evaluated Matrices}}{section.C.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.1}\leavevmode {\color  {Chapter }Eval (for matrices created with HomalgInitialMatrix)}}{72}{subsection.C.4.1}}
\newlabel{Eval:for matrices created with HomalgInitialMatrix}{{C.4.1}{72}{\textcolor {Chapter }{Eval (for matrices created with HomalgInitialMatrix)}}{subsection.C.4.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.2}\leavevmode {\color  {Chapter }Eval (for matrices created with HomalgInitialIdentityMatrix)}}{73}{subsection.C.4.2}}
\newlabel{Eval:for matrices created with HomalgInitialIdentityMatrix}{{C.4.2}{73}{\textcolor {Chapter }{Eval (for matrices created with HomalgInitialIdentityMatrix)}}{subsection.C.4.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.3}\leavevmode {\color  {Chapter }Eval (for matrices created with HomalgZeroMatrix)}}{74}{subsection.C.4.3}}
\newlabel{Eval:for matrices created with HomalgZeroMatrix}{{C.4.3}{74}{\textcolor {Chapter }{Eval (for matrices created with HomalgZeroMatrix)}}{subsection.C.4.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.4}\leavevmode {\color  {Chapter }Eval (for matrices created with HomalgIdentityMatrix)}}{75}{subsection.C.4.4}}
\newlabel{Eval:for matrices created with HomalgIdentityMatrix}{{C.4.4}{75}{\textcolor {Chapter }{Eval (for matrices created with HomalgIdentityMatrix)}}{subsection.C.4.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.5}\leavevmode {\color  {Chapter }Eval (for matrices created with LeftInverseLazy)}}{76}{subsection.C.4.5}}
\newlabel{Eval:for matrices created with LeftInverseLazy}{{C.4.5}{76}{\textcolor {Chapter }{Eval (for matrices created with LeftInverseLazy)}}{subsection.C.4.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.6}\leavevmode {\color  {Chapter }Eval (for matrices created with RightInverseLazy)}}{77}{subsection.C.4.6}}
\newlabel{Eval:for matrices created with RightInverseLazy}{{C.4.6}{77}{\textcolor {Chapter }{Eval (for matrices created with RightInverseLazy)}}{subsection.C.4.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.7}\leavevmode {\color  {Chapter }Eval (for matrices created with Involution)}}{77}{subsection.C.4.7}}
\newlabel{Eval:for matrices created with Involution}{{C.4.7}{77}{\textcolor {Chapter }{Eval (for matrices created with Involution)}}{subsection.C.4.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.8}\leavevmode {\color  {Chapter }Eval (for matrices created with CertainRows)}}{78}{subsection.C.4.8}}
\newlabel{Eval:for matrices created with CertainRows}{{C.4.8}{78}{\textcolor {Chapter }{Eval (for matrices created with CertainRows)}}{subsection.C.4.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.9}\leavevmode {\color  {Chapter }Eval (for matrices created with CertainColumns)}}{79}{subsection.C.4.9}}
\newlabel{Eval:for matrices created with CertainColumns}{{C.4.9}{79}{\textcolor {Chapter }{Eval (for matrices created with CertainColumns)}}{subsection.C.4.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.10}\leavevmode {\color  {Chapter }Eval (for matrices created with UnionOfRows)}}{79}{subsection.C.4.10}}
\newlabel{Eval:for matrices created with UnionOfRows}{{C.4.10}{79}{\textcolor {Chapter }{Eval (for matrices created with UnionOfRows)}}{subsection.C.4.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.11}\leavevmode {\color  {Chapter }Eval (for matrices created with UnionOfColumns)}}{80}{subsection.C.4.11}}
\newlabel{Eval:for matrices created with UnionOfColumns}{{C.4.11}{80}{\textcolor {Chapter }{Eval (for matrices created with UnionOfColumns)}}{subsection.C.4.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.12}\leavevmode {\color  {Chapter }Eval (for matrices created with DiagMat)}}{81}{subsection.C.4.12}}
\newlabel{Eval:for matrices created with DiagMat}{{C.4.12}{81}{\textcolor {Chapter }{Eval (for matrices created with DiagMat)}}{subsection.C.4.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.13}\leavevmode {\color  {Chapter }Eval (for matrices created with KroneckerMat)}}{82}{subsection.C.4.13}}
\newlabel{Eval:for matrices created with KroneckerMat}{{C.4.13}{82}{\textcolor {Chapter }{Eval (for matrices created with KroneckerMat)}}{subsection.C.4.13}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.14}\leavevmode {\color  {Chapter }Eval (for matrices created with MulMat)}}{83}{subsection.C.4.14}}
\newlabel{Eval:for matrices created with MulMat}{{C.4.14}{83}{\textcolor {Chapter }{Eval (for matrices created with MulMat)}}{subsection.C.4.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.15}\leavevmode {\color  {Chapter }Eval (for matrices created with AddMat)}}{84}{subsection.C.4.15}}
\newlabel{Eval:for matrices created with AddMat}{{C.4.15}{84}{\textcolor {Chapter }{Eval (for matrices created with AddMat)}}{subsection.C.4.15}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.16}\leavevmode {\color  {Chapter }Eval (for matrices created with SubMat)}}{84}{subsection.C.4.16}}
\newlabel{Eval:for matrices created with SubMat}{{C.4.16}{84}{\textcolor {Chapter }{Eval (for matrices created with SubMat)}}{subsection.C.4.16}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {C.4.17}\leavevmode {\color  {Chapter }Eval (for matrices created with Compose)}}{85}{subsection.C.4.17}}
\newlabel{Eval:for matrices created with Compose}{{C.4.17}{85}{\textcolor {Chapter }{Eval (for matrices created with Compose)}}{subsection.C.4.17}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {D}\leavevmode {\color  {Chapter }The subpackage \textsf  {ResidueClassRingForHomalg} as a sample ring package}}{87}{appendix.D}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{ResidueClassRingForHomalg}{{D}{87}{\textcolor {Chapter }{The subpackage \textsf {ResidueClassRingForHomalg} as a sample ring package}}{appendix.D}{}}
\@writefile{toc}{\contentsline {section}{\numberline {D.1}\leavevmode {\color  {Chapter }The Mandatory Basic Operations}}{87}{section.D.1}}
\newlabel{ResidueClassRingForHomalg:BasicNoFallBack}{{D.1}{87}{\textcolor {Chapter }{The Mandatory Basic Operations}}{section.D.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.1}\leavevmode {\color  {Chapter }BasisOfRowModule (ResidueClassRing)}}{87}{subsection.D.1.1}}
\newlabel{BasisOfRowModule:ResidueClassRing}{{D.1.1}{87}{\textcolor {Chapter }{BasisOfRowModule (ResidueClassRing)}}{subsection.D.1.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.2}\leavevmode {\color  {Chapter }BasisOfColumnModule (ResidueClassRing)}}{87}{subsection.D.1.2}}
\newlabel{BasisOfColumnModule:ResidueClassRing}{{D.1.2}{87}{\textcolor {Chapter }{BasisOfColumnModule (ResidueClassRing)}}{subsection.D.1.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.3}\leavevmode {\color  {Chapter }DecideZeroRows (ResidueClassRing)}}{88}{subsection.D.1.3}}
\newlabel{DecideZeroRows:ResidueClassRing}{{D.1.3}{88}{\textcolor {Chapter }{DecideZeroRows (ResidueClassRing)}}{subsection.D.1.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.4}\leavevmode {\color  {Chapter }DecideZeroColumns (ResidueClassRing)}}{88}{subsection.D.1.4}}
\newlabel{DecideZeroColumns:ResidueClassRing}{{D.1.4}{88}{\textcolor {Chapter }{DecideZeroColumns (ResidueClassRing)}}{subsection.D.1.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.5}\leavevmode {\color  {Chapter }SyzygiesGeneratorsOfRows (ResidueClassRing)}}{88}{subsection.D.1.5}}
\newlabel{SyzygiesGeneratorsOfRows:ResidueClassRing}{{D.1.5}{88}{\textcolor {Chapter }{SyzygiesGeneratorsOfRows (ResidueClassRing)}}{subsection.D.1.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.6}\leavevmode {\color  {Chapter }SyzygiesGeneratorsOfColumns (ResidueClassRing)}}{89}{subsection.D.1.6}}
\newlabel{SyzygiesGeneratorsOfColumns:ResidueClassRing}{{D.1.6}{89}{\textcolor {Chapter }{SyzygiesGeneratorsOfColumns (ResidueClassRing)}}{subsection.D.1.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.7}\leavevmode {\color  {Chapter }BasisOfRowsCoeff (ResidueClassRing)}}{90}{subsection.D.1.7}}
\newlabel{BasisOfRowsCoeff:ResidueClassRing}{{D.1.7}{90}{\textcolor {Chapter }{BasisOfRowsCoeff (ResidueClassRing)}}{subsection.D.1.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.8}\leavevmode {\color  {Chapter }BasisOfColumnsCoeff (ResidueClassRing)}}{90}{subsection.D.1.8}}
\newlabel{BasisOfColumnsCoeff:ResidueClassRing}{{D.1.8}{90}{\textcolor {Chapter }{BasisOfColumnsCoeff (ResidueClassRing)}}{subsection.D.1.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.9}\leavevmode {\color  {Chapter }DecideZeroRowsEffectively (ResidueClassRing)}}{91}{subsection.D.1.9}}
\newlabel{DecideZeroRowsEffectively:ResidueClassRing}{{D.1.9}{91}{\textcolor {Chapter }{DecideZeroRowsEffectively (ResidueClassRing)}}{subsection.D.1.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.10}\leavevmode {\color  {Chapter }DecideZeroColumnsEffectively (ResidueClassRing)}}{91}{subsection.D.1.10}}
\newlabel{DecideZeroColumnsEffectively:ResidueClassRing}{{D.1.10}{91}{\textcolor {Chapter }{DecideZeroColumnsEffectively (ResidueClassRing)}}{subsection.D.1.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.11}\leavevmode {\color  {Chapter }RelativeSyzygiesGeneratorsOfRows (ResidueClassRing)}}{92}{subsection.D.1.11}}
\newlabel{RelativeSyzygiesGeneratorsOfRows:ResidueClassRing}{{D.1.11}{92}{\textcolor {Chapter }{RelativeSyzygiesGeneratorsOfRows (ResidueClassRing)}}{subsection.D.1.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.1.12}\leavevmode {\color  {Chapter }RelativeSyzygiesGeneratorsOfColumns (ResidueClassRing)}}{92}{subsection.D.1.12}}
\newlabel{RelativeSyzygiesGeneratorsOfColumns:ResidueClassRing}{{D.1.12}{92}{\textcolor {Chapter }{RelativeSyzygiesGeneratorsOfColumns (ResidueClassRing)}}{subsection.D.1.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {D.2}\leavevmode {\color  {Chapter }The Mandatory Tool Operations}}{93}{section.D.2}}
\newlabel{ResidueClassRingForHomalg:ToolsNoFallBack}{{D.2}{93}{\textcolor {Chapter }{The Mandatory Tool Operations}}{section.D.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.1}\leavevmode {\color  {Chapter }InitialMatrix (ResidueClassRing)}}{93}{subsection.D.2.1}}
\newlabel{InitialMatrix:ResidueClassRing}{{D.2.1}{93}{\textcolor {Chapter }{InitialMatrix (ResidueClassRing)}}{subsection.D.2.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.2}\leavevmode {\color  {Chapter }InitialIdentityMatrix (ResidueClassRing)}}{93}{subsection.D.2.2}}
\newlabel{InitialIdentityMatrix:ResidueClassRing}{{D.2.2}{93}{\textcolor {Chapter }{InitialIdentityMatrix (ResidueClassRing)}}{subsection.D.2.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.3}\leavevmode {\color  {Chapter }ZeroMatrix (ResidueClassRing)}}{93}{subsection.D.2.3}}
\newlabel{ZeroMatrix:ResidueClassRing}{{D.2.3}{93}{\textcolor {Chapter }{ZeroMatrix (ResidueClassRing)}}{subsection.D.2.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.4}\leavevmode {\color  {Chapter }IdentityMatrix (ResidueClassRing)}}{94}{subsection.D.2.4}}
\newlabel{IdentityMatrix:ResidueClassRing}{{D.2.4}{94}{\textcolor {Chapter }{IdentityMatrix (ResidueClassRing)}}{subsection.D.2.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.5}\leavevmode {\color  {Chapter }Involution (ResidueClassRing)}}{94}{subsection.D.2.5}}
\newlabel{Involution:ResidueClassRing}{{D.2.5}{94}{\textcolor {Chapter }{Involution (ResidueClassRing)}}{subsection.D.2.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.6}\leavevmode {\color  {Chapter }CertainRows (ResidueClassRing)}}{94}{subsection.D.2.6}}
\newlabel{CertainRows:ResidueClassRing}{{D.2.6}{94}{\textcolor {Chapter }{CertainRows (ResidueClassRing)}}{subsection.D.2.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.7}\leavevmode {\color  {Chapter }CertainColumns (ResidueClassRing)}}{95}{subsection.D.2.7}}
\newlabel{CertainColumns:ResidueClassRing}{{D.2.7}{95}{\textcolor {Chapter }{CertainColumns (ResidueClassRing)}}{subsection.D.2.7}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.8}\leavevmode {\color  {Chapter }UnionOfRows (ResidueClassRing)}}{95}{subsection.D.2.8}}
\newlabel{UnionOfRows:ResidueClassRing}{{D.2.8}{95}{\textcolor {Chapter }{UnionOfRows (ResidueClassRing)}}{subsection.D.2.8}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.9}\leavevmode {\color  {Chapter }UnionOfColumns (ResidueClassRing)}}{96}{subsection.D.2.9}}
\newlabel{UnionOfColumns:ResidueClassRing}{{D.2.9}{96}{\textcolor {Chapter }{UnionOfColumns (ResidueClassRing)}}{subsection.D.2.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.10}\leavevmode {\color  {Chapter }DiagMat (ResidueClassRing)}}{96}{subsection.D.2.10}}
\newlabel{DiagMat:ResidueClassRing}{{D.2.10}{96}{\textcolor {Chapter }{DiagMat (ResidueClassRing)}}{subsection.D.2.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.11}\leavevmode {\color  {Chapter }KroneckerMat (ResidueClassRing)}}{97}{subsection.D.2.11}}
\newlabel{KroneckerMat:ResidueClassRing}{{D.2.11}{97}{\textcolor {Chapter }{KroneckerMat (ResidueClassRing)}}{subsection.D.2.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.12}\leavevmode {\color  {Chapter }MulMat (ResidueClassRing)}}{97}{subsection.D.2.12}}
\newlabel{MulMat:ResidueClassRing}{{D.2.12}{97}{\textcolor {Chapter }{MulMat (ResidueClassRing)}}{subsection.D.2.12}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.13}\leavevmode {\color  {Chapter }AddMat (ResidueClassRing)}}{97}{subsection.D.2.13}}
\newlabel{AddMat:ResidueClassRing}{{D.2.13}{97}{\textcolor {Chapter }{AddMat (ResidueClassRing)}}{subsection.D.2.13}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.14}\leavevmode {\color  {Chapter }SubMat (ResidueClassRing)}}{98}{subsection.D.2.14}}
\newlabel{SubMat:ResidueClassRing}{{D.2.14}{98}{\textcolor {Chapter }{SubMat (ResidueClassRing)}}{subsection.D.2.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.15}\leavevmode {\color  {Chapter }Compose (ResidueClassRing)}}{98}{subsection.D.2.15}}
\newlabel{Compose:ResidueClassRing}{{D.2.15}{98}{\textcolor {Chapter }{Compose (ResidueClassRing)}}{subsection.D.2.15}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.16}\leavevmode {\color  {Chapter }IsZeroMatrix (ResidueClassRing)}}{98}{subsection.D.2.16}}
\newlabel{IsZeroMatrix:ResidueClassRing}{{D.2.16}{98}{\textcolor {Chapter }{IsZeroMatrix (ResidueClassRing)}}{subsection.D.2.16}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.17}\leavevmode {\color  {Chapter }NrRows (ResidueClassRing)}}{98}{subsection.D.2.17}}
\newlabel{NrRows:ResidueClassRing}{{D.2.17}{98}{\textcolor {Chapter }{NrRows (ResidueClassRing)}}{subsection.D.2.17}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.18}\leavevmode {\color  {Chapter }NrColumns (ResidueClassRing)}}{98}{subsection.D.2.18}}
\newlabel{NrColumns:ResidueClassRing}{{D.2.18}{98}{\textcolor {Chapter }{NrColumns (ResidueClassRing)}}{subsection.D.2.18}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.2.19}\leavevmode {\color  {Chapter }Determinant (ResidueClassRing)}}{99}{subsection.D.2.19}}
\newlabel{Determinant:ResidueClassRing}{{D.2.19}{99}{\textcolor {Chapter }{Determinant (ResidueClassRing)}}{subsection.D.2.19}{}}
\@writefile{toc}{\contentsline {section}{\numberline {D.3}\leavevmode {\color  {Chapter }Some of the Recommended Tool Operations}}{99}{section.D.3}}
\newlabel{ResidueClassRingForHomalg:ToolsFallBack}{{D.3}{99}{\textcolor {Chapter }{Some of the Recommended Tool Operations}}{section.D.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.3.1}\leavevmode {\color  {Chapter }AreEqualMatrices (ResidueClassRing)}}{99}{subsection.D.3.1}}
\newlabel{AreEqualMatrices:ResidueClassRing}{{D.3.1}{99}{\textcolor {Chapter }{AreEqualMatrices (ResidueClassRing)}}{subsection.D.3.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.3.2}\leavevmode {\color  {Chapter }IsOne (ResidueClassRing)}}{99}{subsection.D.3.2}}
\newlabel{IsOne:ResidueClassRing}{{D.3.2}{99}{\textcolor {Chapter }{IsOne (ResidueClassRing)}}{subsection.D.3.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.3.3}\leavevmode {\color  {Chapter }IsDiagonalMatrix (ResidueClassRing)}}{99}{subsection.D.3.3}}
\newlabel{IsDiagonalMatrix:ResidueClassRing}{{D.3.3}{99}{\textcolor {Chapter }{IsDiagonalMatrix (ResidueClassRing)}}{subsection.D.3.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.3.4}\leavevmode {\color  {Chapter }ZeroRows (ResidueClassRing)}}{100}{subsection.D.3.4}}
\newlabel{ZeroRows:ResidueClassRing}{{D.3.4}{100}{\textcolor {Chapter }{ZeroRows (ResidueClassRing)}}{subsection.D.3.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {D.3.5}\leavevmode {\color  {Chapter }ZeroColumns (ResidueClassRing)}}{100}{subsection.D.3.5}}
\newlabel{ZeroColumns:ResidueClassRing}{{D.3.5}{100}{\textcolor {Chapter }{ZeroColumns (ResidueClassRing)}}{subsection.D.3.5}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {E}\leavevmode {\color  {Chapter }Debugging \textsf  {MatricesForHomalg}}}{101}{appendix.E}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{Debugging}{{E}{101}{\textcolor {Chapter }{Debugging \textsf {MatricesForHomalg}}}{appendix.E}{}}
\@writefile{toc}{\contentsline {section}{\numberline {E.1}\leavevmode {\color  {Chapter }Increase the assertion level}}{101}{section.E.1}}
\newlabel{SetAssertionLevel}{{E.1}{101}{\textcolor {Chapter }{Increase the assertion level}}{section.E.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {E.2}\leavevmode {\color  {Chapter }\texttt  {Using homalgMode}}}{103}{section.E.2}}
\newlabel{using homalgMode}{{E.2}{103}{\textcolor {Chapter }{\texttt {Using homalgMode}}}{section.E.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {E.2.1}\leavevmode {\color  {Chapter }homalgMode}}{103}{subsection.E.2.1}}
\newlabel{homalgMode}{{E.2.1}{103}{\textcolor {Chapter }{homalgMode}}{subsection.E.2.1}{}}
\@writefile{toc}{\contentsline {chapter}{\numberline {F}\leavevmode {\color  {Chapter }Overview of the \textsf  {MatricesForHomalg} Package Source Code}}{105}{appendix.F}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{FileOverview}{{F}{105}{\textcolor {Chapter }{Overview of the \textsf {MatricesForHomalg} Package Source Code}}{appendix.F}{}}
\@writefile{toc}{\contentsline {section}{\numberline {F.1}\leavevmode {\color  {Chapter }Rings, Ring Maps, Matrices, Ring Relations}}{105}{section.F.1}}
\newlabel{RingsMapsMatrices}{{F.1}{105}{\textcolor {Chapter }{Rings, Ring Maps, Matrices, Ring Relations}}{section.F.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {F.2}\leavevmode {\color  {Chapter }The Low Level Algorithms}}{105}{section.F.2}}
\newlabel{Low Level Algorithms}{{F.2}{105}{\textcolor {Chapter }{The Low Level Algorithms}}{section.F.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {F.3}\leavevmode {\color  {Chapter }Logical Implications for \textsf  {MatricesForHomalg} Objects}}{106}{section.F.3}}
\newlabel{Logical Implications}{{F.3}{106}{\textcolor {Chapter }{Logical Implications for \textsf {MatricesForHomalg} Objects}}{section.F.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {F.4}\leavevmode {\color  {Chapter }The subpackage \textsf  {ResidueClassRingForHomalg}}}{106}{section.F.4}}
\newlabel{ResidueClassRingForHomalg_subpackage}{{F.4}{106}{\textcolor {Chapter }{The subpackage \textsf {ResidueClassRingForHomalg}}}{section.F.4}{}}
\bibstyle{alpha}
\bibdata{MatricesForHomalgBib.xml}
\@writefile{toc}{\contentsline {section}{\numberline {F.5}\leavevmode {\color  {Chapter }The homalgTable for \textsf  {GAP4} built-in rings}}{107}{section.F.5}}
\newlabel{The built-in rings}{{F.5}{107}{\textcolor {Chapter }{The homalgTable for \textsf {GAP4} built-in rings}}{section.F.5}{}}
\bibcite{BR}{BR08}
\bibcite{McCRob}{MR01}
\@writefile{toc}{\contentsline {chapter}{References}{108}{appendix*.3}}
\@writefile{toc}{\contentsline {chapter}{Index}{109}{section*.4}}
