nLab
commutative loop

Contents

Contents

Definition

With multiplication, division, and identity

A commutative loop is a commutative unital magma (G,()():G×GG,1:G)(G,(-)\cdot(-):G \times G \to G,1:G) equipped with a binary operation ()/():G×GG(-)/(-):G \times G \to G called division such that (x/y)y=x(x/y) \cdot y = x and (xy)/y=x(x \cdot y)/y = x.

With division and identity

A commutative loop is a pointed magma (G,/,1)(G,/,1) such that:

  • For all aa in GG, a/a=1a/a=1
  • For all aa in GG, 1/(1/a)=a1/(1/a)=a
  • For all aa and bb in GG, a/(1/b)=b/(1/a)a/(1/b) = b/(1/a)

with multiplication defined as ab=a/(1/b)a \cdot b= a/(1/b).

With multiplication, inverses, and identity

A commutative loop is a commutative unital magma (G,()():G×GG,1:G)(G, (-)\cdot(-):G\times G\to G,1:G) equipped with a inverse () 1:GG(-)^{-1}:G \to G such that (xy 1)y=x(x \cdot y^{-1}) \cdot y = x and (xy)y 1=x(x \cdot y) \cdot y^{-1} = x.

Examples

Created on May 24, 2021 at 16:43:15. See the history of this page for a list of all contributions to it.