Quasigroup

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QUICK PHRASES: group without identity and associativity, magma with unique left and right quotients

Definition

Definition with symbols

A quasigroup is a magma (set with binary operation) $(S,*)$ with the following properties:

• For every $a,b \in S$, there is a unique $x \in S$ such that $a * x = b$
• For every $a,b \in S$, there is a unique $y \in S$ such that $y * a = b$

Note that $x,y$ may be equal or different for a particular choice of $a$ and $b$.

Equivalently, a quasigroup is a magma where every element is cancellative and where every element is left-accessible and right-accessible from every other element.

Relation with other notions

Stronger notions

Any nonempty quasigroup that is also a semigroup is a group. For full proof, refer: Associative quasigroup implies group

Weaker notions

A finite magma is a quasigroup if and only if its multiplication table is a Latin square.