Definition:Semigroup

Definition
Let $$\left({S, \circ}\right)$$ be a groupoid.

Then $$\left({S, \circ}\right)$$ is a semigroup iff $$\circ$$ is associative on $$S$$.

That is, a semigroup is an algebraic structure which is closed and associative.

Some older texts have this as semi-group.

Warning
Some sources call this a monoid, but this term usually has a more precise meaning.

Make sure you understand which is being used.