nLab
invertible quasigroup

Contents

Idea

A quasigroup with a two-sided inverse

Definition

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

for all a,bGa,b \in G.

Without division

An invertible quasigroup 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

and

for all a,bGa,b \in G.

Examples