\contentsline {chapter}{\numberline {1}Introduction}{4}
\contentsline {section}{\numberline {1.1}Installation}{4}
\contentsline {subsection}{\numberline {1.1.1}Brief description of \texttt {LOOPS}\ files}{4}
\contentsline {section}{\numberline {1.2}Documentation}{4}
\contentsline {section}{\numberline {1.3}Test files}{5}
\contentsline {section}{\numberline {1.4}Mathematical background}{5}
\contentsline {subsection}{\numberline {1.4.1}Quasigroups and loops}{6}
\contentsline {subsection}{\numberline {1.4.2}Translations}{6}
\contentsline {subsection}{\numberline {1.4.3}Homomorphisms and homotopisms}{6}
\contentsline {section}{\numberline {1.5}How the package works}{7}
\contentsline {subsection}{\numberline {1.5.1}Representing quasigroups in \texttt {LOOPS}}{7}
\contentsline {subsection}{\numberline {1.5.2}Calculating with quasigroups in \texttt {LOOPS}}{7}
\contentsline {subsection}{\numberline {1.5.3}Magmas, quasigroups, loops and groups in \texttt {GAP}}{7}
\contentsline {section}{\numberline {1.6}Feedback}{8}
\contentsline {chapter}{\numberline {2}Core methods}{9}
\contentsline {section}{\numberline {2.1}Naming, viewing and printing objects in \texttt {LOOPS}}{9}
\contentsline {subsection}{\numberline {2.1.1}Named quasigroups and loops}{9}
\contentsline {subsection}{\numberline {2.1.2}View mode}{9}
\contentsline {subsection}{\numberline {2.1.3}Print mode}{10}
\contentsline {section}{\numberline {2.2}Creating quasigroups and loops}{10}
\contentsline {subsection}{\numberline {2.2.1}Testing multiplication tables}{10}
\contentsline {subsection}{\numberline {2.2.2}Creating quasigroups and loops manually}{11}
\contentsline {subsection}{\numberline {2.2.3}Creating quasigroups and loops from a file}{11}
\contentsline {subsection}{\numberline {2.2.4}Conversions}{12}
\contentsline {subsection}{\numberline {2.2.5}Products of loops}{13}
\contentsline {subsection}{\numberline {2.2.6}Opposite quasigroups and loops}{13}
\contentsline {section}{\numberline {2.3}GAP categories}{13}
\contentsline {section}{\numberline {2.4}Basic attributes}{13}
\contentsline {section}{\numberline {2.5}Basic arithmetic operations}{14}
\contentsline {subsection}{\numberline {2.5.1}Multiplication}{14}
\contentsline {subsection}{\numberline {2.5.2}Division}{14}
\contentsline {subsection}{\numberline {2.5.3}Powers and inverses}{15}
\contentsline {subsection}{\numberline {2.5.4}Associators and commutators}{15}
\contentsline {section}{\numberline {2.6}Generators}{15}
\contentsline {section}{\numberline {2.7}Permutations associated with loops}{16}
\contentsline {subsection}{\numberline {2.7.1}Sections}{16}
\contentsline {subsection}{\numberline {2.7.2}Multiplication groups}{16}
\contentsline {subsection}{\numberline {2.7.3}Inner mapping group}{16}
\contentsline {section}{\numberline {2.8}Subquasigroups and subloops}{17}
\contentsline {section}{\numberline {2.9}Nucleus, commutant, center}{18}
\contentsline {section}{\numberline {2.10}Testing properties}{18}
\contentsline {subsection}{\numberline {2.10.1}Associativity, commutativity and generalizations}{19}
\contentsline {subsection}{\numberline {2.10.2}Inverse properties}{19}
\contentsline {subsection}{\numberline {2.10.3}Properties of quasigroups}{19}
\contentsline {subsection}{\numberline {2.10.4}Loops of Bol-Moufang type and related properties}{20}
\contentsline {subsection}{\numberline {2.10.5}Conjugacy closed loops}{22}
\contentsline {subsection}{\numberline {2.10.6}Additional varieties of loops}{22}
\contentsline {section}{\numberline {2.11}Normality}{22}
\contentsline {section}{\numberline {2.12}Factor loop}{23}
\contentsline {section}{\numberline {2.13}Nilpotency}{23}
\contentsline {section}{\numberline {2.14}Solvability}{23}
\contentsline {section}{\numberline {2.15}Filters built into \texttt {LOOPS}}{24}
\contentsline {chapter}{\numberline {3}Specific methods}{25}
\contentsline {section}{\numberline {3.1}Isomorphisms and automorphisms}{25}
\contentsline {subsection}{\numberline {3.1.1}Discriminator}{25}
\contentsline {section}{\numberline {3.2}Moufang modifications}{26}
\contentsline {subsection}{\numberline {3.2.1}Cyclic modification}{26}
\contentsline {subsection}{\numberline {3.2.2}Dihedral modification}{26}
\contentsline {subsection}{\numberline {3.2.3}Loops $M(G,2)$}{26}
\contentsline {section}{\numberline {3.3}Triality for Moufang loops}{27}
\contentsline {chapter}{\numberline {4}Libraries of small loops}{28}
\contentsline {section}{\numberline {4.1}A typical library}{28}
\contentsline {section}{\numberline {4.2}Left Bol loops}{28}
\contentsline {section}{\numberline {4.3}Small Moufang loops}{29}
\contentsline {subsection}{\numberline {4.3.1}Search for additional Moufang loops}{29}
\contentsline {section}{\numberline {4.4}Steiner loops}{29}
\contentsline {section}{\numberline {4.5}Paige loops}{30}
\contentsline {section}{\numberline {4.6}Interesting loops}{30}
\contentsline {chapter}{\numberline {5}Plans for future versions}{31}
\contentsline {section}{\numberline {5.1}Alternative representations of quasigroups and loops in \texttt {GAP}}{31}
\contentsline {section}{\numberline {5.2}Better support for quasigroups}{31}
\contentsline {section}{\numberline {5.3}More homomorphisms and homotopisms}{31}
\contentsline {section}{\numberline {5.4}Expanded libraries}{31}
\contentsline {section}{\numberline {5.5}Bits and pieces}{32}
\contentsline {chapter}{Bibliography}{33}
\contentsline {chapter}{Index}{34}
