nLab
invertible quasigroup

Contents

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

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

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

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

and

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

for all a,bGa,b \in G.

Examples

Last revised on June 14, 2021 at 10:58:17. See the history of this page for a list of all contributions to it.