Definition:Quasigroup

Definition
A quasigroup is a magma $\left({S, \circ}\right)$ 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:
 * $\forall a, b \in S: \exists ! x: x \circ a = b$
 * $\forall a, b \in S: \exists ! y: a \circ y = b$