# Definition:Quasigroup

## Definition

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:

$\forall a, b \in S: \exists ! x: x \circ a = b$
$\forall a, b \in S: \exists ! y: a \circ y = b$

### Left Quasigroup

$\left({S, \circ}\right)$ is a left quasigroup if and only if:

for all $a \in S$, the left regular representation $\lambda_a$ is a permutations on $S$.

That is:

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

### Right Quasigroup

$\left({S, \circ}\right)$ is a right quasigroup if and only if:

for all $a \in S$, the right regular representation $\rho_a$ is a permutations on $S$.

That is:

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

## Also see

• Results about quasigroups can be found here.