Definition:Quasigroup/Left Quasigroup

Definition
Let $\left({S, \circ}\right)$ be a magma.

$\left({S, \circ}\right)$ is a left quasigroup iff:
 * for all $a \in S$, the left regular representation $\lambda_a$ is a permutations on $S$.

That is:
 * $\forall a, b \in S: \exists ! x: a \circ x = b$

Also see

 * Definition:Right Quasigroup