In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under function composition.

Commutative semigroup with associative operation.

(mathematics) a function that preserves semigroup structure

(algebra, group theory) A semigroup in which every element x has an inverse y, such that x = xyx and y = yxy.

In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations.