\begin{definition}[Definition:Quasigroup]
A '''quasigroup''' is a magma $\struct {S, \circ}$ which has the Latin square property.
That is, such that $\forall a \in S$, the left and right regular representations $\lambda_a$ and $\rho_a$ are permutations on $S$.
That is:
:$\forall a, b \in S: \exists ! x \in S: x \circ a = b$
:$\forall a, b \in S: \exists ! y \in S: a \circ y = b$
\end{definition}