Definition:Quasigroup
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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:
- $\forall a, b \in S: \exists ! x \in S: x \circ a = b$
- $\forall a, b \in S: \exists ! y \in S: a \circ y = b$
Left Quasigroup
$\struct {S, \circ}$ 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:
- $\forall a, b \in S: \exists ! x \in S: a \circ x = b$
Right Quasigroup
$\struct {S, \circ}$ is a right quasigroup if and only if:
- for all $a \in S$, the right regular representation $\rho_a$ is a permutation on $S$.
That is:
- $\forall a, b \in S: \exists ! x \in S: x \circ a = b$
Also see
- Results about quasigroups can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.8$