Definition:Semigroup

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

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.