semigroup

A semigroup is an algebraic structure consisting of a set equipped with an associative binary operation.

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

1. $S$ is closed: $ab\in S$
2. $S$ is associative: $(ab)c=a(bc)$

A semigroup is an associative magma.
A semigroup with an identity element is a monoid.