nLab
invertible semigroup

This entry is about semigroups with a two-sided inverse. For the semigroup with a unary operator ii such that si(s)s=ss \cdot i(s) \cdot s = s and i(s)si(s)=i(s)i(s) \cdot s \cdot i(s) = i(s), see inverse semigroup.

Context

Group Theory

Representation theory

Contents

Idea

A semigroup that is also an invertible magma.

Definition

With multiplication and inverses

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

for all a,bGa,b \in G.

Torsor-like definition

There is an alternate definition of an invertible semigroup that looks like the usual definition of a torsor or heap:

An invertible semigroup is a set SS with a binary operation ()():S×SS(-)\cdot(-):S\times S\to S called multiplication and a unary operation () 1:SS(-)^{-1}:S\to S called inverse satisfying the following laws:

Pseudo-torsor

Every invertible semigroup GG has a pseudo-torsor, or associative Malcev algebras, t:G 3Gt:G^3\to G defined as t(x,y,z)=xy 1zt(x,y,z)=x\cdot y^{-1}\cdot z. If the invertible semigroup is inhabited, then those pseudo-torsors are actually torsors or heaps.

Properties