nLab
invertible magma

Contents

Contents

Idea

It is possible to define inverses in a magma without defining an identity element first, yielding a notion of invertible magma

Definition

A left 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 left inverse or retraction such that

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

for all a,bGa,b \in G.

A right 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 right inverse or section such that

  • 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.

An 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(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.

Properties

Every invertible magma is a cancellative magma?.

The submagma of every power-associative invertible magma MM generated by an element aMa \in M is a cyclic group. This means in particular there is a \mathbb{Z}-action on MM () ():M×M(-)^{(-)}:M\times\mathbb{Z}\to M called the power.

Examples

Last revised on June 4, 2021 at 21:51:02. See the history of this page for a list of all contributions to it.