In mathematics, a ****transformation, transform, or self-map is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. Examples include linear ****transformations of vector spaces and geometric ****transformations, which include projective ****transformations, affine ****transformations, and specific affine ****transformations, such as rotations, reflections and translations.

Partial transformations

While it is common to use the term transformation for any function of a set into itself (especially in terms like "transformation semigroup" and similar), there exists an alternative form of terminological convention in which the term "transformation" is reserved only for bijections. When such a narrow notion of transformation is generalized to partial functions, then a partial transformation is a function f: A → B, where both A and B are subsets of some set X.

Algebraic structures

The set of all transformations on a given base set, together with function composition, forms a regular semigroup.

Combinatorics

For a finite set of cardinality n, there are nn transformations and (n+1)n partial transformations.