Loops

http://mathcs.chapman.edu/structuresold/files/Loops.pdf

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\begin{document}
\textbf{\Large Loops}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Loops}{edit}

\abbreviation{Loop}
\begin{definition}
A \emph{loop} is a structure $\mathbf{A}=\langle A,\cdot ,\backslash,/,e\rangle $ of type $\langle 2,2,2,0\rangle $ such that

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

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

$e$ is an identity for $\cdot$:  $xe = x$, $ex = x$

Remark: 
\end{definition}

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

$h(xy)=h(x)h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$
\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 $n$.

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\end{finite_members}
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\begin{subclasses}\ 

\href{Groups.pdf}{Groups} 

\end{subclasses}
\begin{superclasses}\ 

\href{Quasigroups.pdf}{Quasigroups} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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