Semigroup
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 generalized a group in that no element had to have an inverse, thus the name
semigroup. The binary operation of a semigroup is most often denoted multiplicatively:, or simply, denotes the result of applying the semigroup operation to the ordered pair. The operation is required to be associative so that for all
x,
y and
z, but need not be commutative so that does not have to equal . 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. The formal study of semigroups began in the early 20th century.