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.

Some sources present the monoid as $\struct {S, \circ, e_S}$ in order to place emphasis on the identity element, but this approach is not endorsed on as it complicates notation to little advantage.

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.