Transformation (function)

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title: "Transformation (function)" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["transformation-(function)", "functions-and-mappings"] description: "Function that applies a set to itself" topic_path: "general/transformation-function" source: "https://en.wikipedia.org/wiki/Transformation_(function)" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Function that applies a set to itself ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/9/9c/A_code_snippet_for_a_rhombic_repetitive_pattern.svg" caption="linear]]."] ::

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 n**n transformations and (n+1)n partial transformations.

References

References

1. "Self-Map -- from Wolfram MathWorld".
2. (2008). "Classical Finite Transformation Semigroups: An Introduction". Springer Science & Business Media.
3. Pierre A. Grillet. (1995). "Semigroups: An Introduction to the Structure Theory". CRC Press.
4. Wilkinson, Leland. (2005). "The Grammar of Graphics". Springer.
5. "Transformations".
6. "Types of Transformations in Math".
7. Christopher Hollings. (2014). "Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups". American Mathematical Society.
8. (2008). "Classical Finite Transformation Semigroups: An Introduction". Springer Science & Business Media.

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