Definition:Monoid

Definition
A semigroup with an identity element is called a monoid.

That is, a monoid is an algebraic structure $\left({S, \circ, e_S}\right)$ which satisfies the following three properties:

The element $e_S$ is called the identity.

From Identity of Semigroup is Unique, there can only be one such identity element.

A monoid can not be empty, because it must at least have an identity.

Also see

 * Monoid Category, modelling monoids in category theory.