nLab commutative invertible quasigroup

Idea

There should be a commutative version of a invertible quasigroup or an invertible version of a commutative quasigroup. That leads to the concept of a commutative invertible quasigroup.

Definition

A commutative invertible quasigroup is a commutative quasigroup $(G,\cdot,/)$ with a unary operation $(-)^{-1}:G \to G$ called the inverse such that

• $a \cdot (b \cdot b^{-1}) = a$

for all $a,b \in G$.

Without division

A commutative invertible quasigroup is a commutative magma $(G,(-)\cdot(-):G\times G\to G)$ with a unary operation $(-)^{-1}:G \to G$ called the inverse such that

• $a \cdot (b \cdot b^{-1}) = a$
• $(a \cdot b) \cdot b^{-1} = a$
• $(a \cdot b^{-1}) \cdot b = a$

for all $a,b \in G$.

Examples

• Every commutative loop is a commutative invertible unital quasigroup.

• Every commutative invertible semigroup is a commutative associative quasigroup.

• Every abelian group is a commutative invertible monoid.

• The empty quasigroup is a commutative invertible quasigroup.

Related concepts

• invertible quasigroup (non-commutative version)

• commutative quasigroup (non-invertible version)

• commutative loop (unital version)

• commutative invertible semigroup (associative version)