Transformation (function)

In mathematics, a transformation, transform, or self-map^[1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X.^[2]^[3]^[4] 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.^[5]^[6]

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.^[7]

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.^[8]

References

^ "Self-Map -- from Wolfram MathWorld". Retrieved March 4, 2024.
^ Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 1. ISBN 978-1-84800-281-4.
^ Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4.
^ Wilkinson, Leland (2005). The Grammar of Graphics (2nd ed.). Springer. p. 29. ISBN 978-0-387-24544-7.
^ "Transformations". www.mathsisfun.com. Retrieved 2019-12-13.
^ "Types of Transformations in Math". Basic-mathematics.com. Retrieved 2019-12-13.
^ Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.
^ Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 2. ISBN 978-1-84800-281-4.

External links

Media related to Transformation (function) at Wikimedia Commons

[1] "Self-Map -- from Wolfram MathWorld". Retrieved March 4, 2024.

[2] Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 1. ISBN 978-1-84800-281-4.

[Grillet1995-3] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4.

[4] Wilkinson, Leland (2005). The Grammar of Graphics (2nd ed.). Springer. p. 29. ISBN 978-0-387-24544-7.

[5] "Transformations". www.mathsisfun.com. Retrieved 2019-12-13.

[:0-6] "Types of Transformations in Math". Basic-mathematics.com. Retrieved 2019-12-13.

[Hollings2014-7] Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.

[8] Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 2. ISBN 978-1-84800-281-4.

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