nLab commutative quasigroup

Definition

A commutative quasigroup is a commutative magma $(G,(-)\cdot(-):G \times G \to G)$ equipped with a binary operation $(-)/(-):G \times G \to G$ called division such that $(x/y) \cdot y = x$ and $(x \cdot y)/y = x$.

Examples

• Every commutative loop is a commutative quasigroup.

• Every commutative invertible quasigroup is a commutative quasigroup.

• The empty quasigroup is a commutative quasigroup.

Related concepts

• associative quasigroup