nLab
commutative invertible magma

Contents

Contents

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,()():G×GG)(G,(-)\cdot(-):G\times G\to G) with a unary operation () 1:GG(-)^{-1}:G \to G called the inverse such that

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

for all a,bGa,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.

Last revised on May 25, 2021 at 10:26:44. See the history of this page for a list of all contributions to it.