Mathematical Structures: Quasigroups

# Quasigroups

Difference (from prior major revision) (author diff)

Changed: 32c32
 A quasigroup is a structure $\mathbf{A}=\langle A,\cdot ,\backslash,/\rangle$ of type $\langle 2,2,2\rangle$ such that
 A quasigroup is a structure $\mathbf{A}=\langle A,\cdot ,\backslash,/\rangle$ of type $\langle 2,2,2\rangle$ such that

Changed: 35c35
 $(y/x)x = y$, $x(x\y) = y$
 $(y/x)x = y$, $x(x\backslash y) = y$

Changed: 38c38
 $(xy)/y = x$, $x\(xy) = y$
 $(xy)/y = x$, $x\backslash(xy) = y$

Changed: 47,48c47
 $h(xy)=h(x)h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$
 $h(xy)=h(x)h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$.

 \href{Medial_quasigroups.pdf}{Medial quasigroups}

http://mathcs.chapman.edu/structuresold/files/Quasigroups.pdf
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\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
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\begin{document}
\textbf{\Large Quasigroups}

\abbreviation{Qgrp}
\begin{definition}
A \emph{quasigroup} is a structure $\mathbf{A}=\langle A,\cdot ,\backslash,/\rangle$ of type $\langle 2,2,2\rangle$ such that

$(y/x)x = y$, $x(x\backslash y) = y$

$(xy)/y = x$, $x\backslash(xy) = y$

Remark:

\end{definition}
\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be quasigroups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(xy)=h(x)h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$.
\end{morphisms}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & \\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & no\\\hline
Congruence modular & \\\hline
Congruence n-permutable & \\\hline
Congruence regular & \\\hline
Congruence uniform & \\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &5\\ f(5)= &35\\ f(6)= &1411\\ f(7)= &1130531\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\href{Loops.pdf}{Loops}

\href{Medial_quasigroups.pdf}{Medial quasigroups}

\end{subclasses}

\begin{superclasses}\

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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