A monoid is a set that is closed under an associative binary operation and has an identity element I element S such that for all a element S, I a = a I = a. Note that unlike a group, its elements need not have inverses. It can also be thought of as a semigroup with an identity element. A monoid must contain at least one element. A monoid that is commutative is, not surprisingly, known as a commutative monoid.

binary operator | commutative monoid | free idempotent monoid | group | semigroup | submonoid