adjoining an identity to a semigroup

It is possible to formally adjoin an identity elementMathworldPlanetmath to any semigroup to make it into a monoid.

Suppose S is a semigroup without an identityPlanetmathPlanetmathPlanetmath, and consider the set S{1} where 1 is a symbol not in S. Extend the semigroup operationMathworldPlanetmath from S to S{1} by additionally defining:

s1=s=1s,for allsS1

It is easy to verify that this defines a semigroup (associativity is the only thing that needs to be checked).

As a matter of notation, it is customary to write S1 for the semigroup S with an identity adjoined in this manner, if S does not already have one, and to agree that S1=S, if S does already have an identity.

Despite the simplicity of this construction, however, it rarely allows one to simplify a problem by considering monoids instead of semigroups. As soon as one starts to look at the structureMathworldPlanetmath of the semigroup, it is almost invariably the case that one needs to consider subsemigroups and ideals of the semigroup which do not contain the identity.

Title adjoining an identity to a semigroup
Canonical name AdjoiningAnIdentityToASemigroup
Date of creation 2013-03-22 13:01:19
Last modified on 2013-03-22 13:01:19
Owner mclase (549)
Last modified by mclase (549)
Numerical id 5
Author mclase (549)
Entry type Definition
Classification msc 20M99
Related topic Semigroup
Related topic Monoid