Quasigroup

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.

Stronger notions

 * Weaker than::Group:
 * Weaker than::Loop: A loop is a quasigroup with a two-sided multiplicative identity element (or neutral element)

Any nonempty quasigroup that is also a semigroup is a group.

Weaker notions

 * Stronger than::Cancellative magma

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