monoid

A monoid is an algebraic structure consisting of a set equipped with an associative binary operation and an identity element.

Formally,
A monoid is a set $M$ equipped with a binary operation such that for all elements $a,b,c\in M$,

1. $M$ is closed: $ab\in M$
2. $M$ is associative: $(ab)c=a(bc)$
3. $M$ has an identity element: $ae=a=ea$

A monoid is a semigroup with an identity element.
A monoid with inverse elements is a group.