nLab commutative invertible magma

Idea

There should be a commutative version of a invertible magma. That leads to the concept of a commutative invertible magma.

Definition

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

• $a \cdot (b \cdot b^{-1}) = a$

for all $a,b \in G$.

Examples

• Every commutative loop is a commutative invertible unital magma.

• Every commutative invertible quasigroup is a commutative invertible magma.

• Every abelian group is a commutative invertible monoid.

• The empty magma is a commutative invertible magma.

Related concepts

• invertible magma (non-commutative version)

• commutative magma (non-invertible version)

• commutative loop (unital version)

• commutative invertible quasigroup (divisible version)

• commutative invertible semigroup (associative version)