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commutative invertible quasigroup

Contents

Contents

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,,/)(G,\cdot,/) 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.

Without division

A commutative invertible quasigroup is a commutative 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
  • (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

  • 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.

Created on May 25, 2021 at 10:16:32. See the history of this page for a list of all contributions to it.