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\import{algebra/magma.tex}
\section{Quasigroups}
\begin{struct}\label{quasigroup}
A quasigroup $A$ is a magma equipped with
\begin{enumerate}
\item $\ldiv$
\item $\rdiv$
\end{enumerate}
such that
\begin{enumerate}
\item for all $a, b\in A$ we have $\ldiv (a,b)\in A$.
\item for all $a, b\in A$ we have $\rdiv (a,b)\in A$.
\item for all $a,b \in A$ we have $b = \mul(a,\ldiv (a,b))$.
\item for all $a,b \in A$ we have $b = \ldiv(a,\mul (a,b))$.
\item for all $a,b \in A$ we have $b = \mul(\rdiv (b,a),a)$.
\item for all $a,b \in A$ we have $b = \rdiv(\mul (b,a),a)$.
\end{enumerate}
\end{struct}
% Cancelling an element on the left.
\begin{lemma}\label{quasigroup_cancel_left}
Let $A$ be a quasigroup.
Let $a,b,c \in A$.
Suppose $\mul(a,b) = \mul(a,c)$.
Then $b = c$.
\end{lemma}
% Cancelling an element on the right.
\begin{lemma}\label{quasigroup_cancel_right}
Let $A$ be a quasigroup.
Let $a,b,c \in A$.
Suppose $\mul(a,c) = \mul(b,c)$.
Then $a = b$.
\end{lemma}
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