Definition:Monoid

Definition
A monoid is a semigroup with an identity element.

Monoid Axioms
The properties that define a monoid can be gathered together as follows:

Also known as
Some treatments of group theory and abstract algebra do not introduce the term monoid, but simply discuss semigroups which happen to have an identity element.

Also see

 * Definition:Group
 * Monoid is not Empty, because it must at least have an identity element.
 * Identity of Monoid is Unique: in a monoid there is only one identity element.


 * Definition:Monoid Category, modelling monoids in category theory.