Definition:Semigroup

Definition
Let $\struct {S, \circ}$ be a magma.

Then $\struct {S, \circ}$ is a semigroup $\circ$ is associative on $S$.

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

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

Also defined as
Some sources specify that a semigroup must be non-empty, thus denying the possibility of $S = \O$ for such a structure.

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

A semigroup is also known as an associative algebraic structure.

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

Make sure you understand which is being used.

Also see

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


 * Definition:Monoid
 * Definition:Group