Definition:Quasigroup/Left Quasigroup

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Let $\struct {S, \circ}$ be a magma.

$\struct {S, \circ}$ is a left quasigroup if and only if:

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

That is:

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

Also see

  • Results about quasigroups can be found here.