Semigroup is an algebraic structure consisting of a set together with an associative binary operation
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that a semigroup need not have an identity element. It also (originally) generalized a group (a monoid with all inverses) to a type where every element did not have to have an inverse, thus the name semigroup.
The binary operation of a semigroup is most often denoted multiplicatively: x•y, or simply xy, denotes the result of applying the semigroup operation to the ordered pair (x,y). The operation is required to be associative so that (x•y)•z = x•(y•z) for all x, y and z, but need not be commutative so that x•y does not have to equal y•x (contrast to the standard multiplication operator on real numbers, where xy = yx).
By definition, a semigroup is an associative magma. A semigroup with an identity element is called a monoid. A group is then a monoid in which every element has an inverse element. Semigroups must not be confused with quasigroups which are sets with a not necessarily associative binary operation such that division is always possible.
Wikipedia page: http://en.wikipedia.org/wiki/Semigroup
