Definition:Semigroup

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

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

That is:
 * A semigroup is an algebraic structure which is closed and associative.

Semigroup Axioms
The properties that define a semigroup can be gathered together as follows:

Also known as
Some older texts have this as semi-group.

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

Make sure you understand which is being used.

Also see

 * Definition:Commutative Semigroup
 * Definition:Ordered Semigroup
 * Definition:Subsemigroup


 * Definition:Monoid
 * Definition:Group