\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}