A semigroup consisting of a collection C of transformations of a set S into itself (see function), the dyadic operation ∘ being the composition of functions; it is essential that the set C should be closed with respect to composition, i.e. if c1 and c2 are in C then so is c1 ∘ c2.
If the identity transformation (see identity function) is included in the transformation semigroup, a transformation monoid results. Every monoid is isomorphic to a transformation monoid.