Transformation (function)

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File:A code snippet for a rhombic repetitive pattern.svg of four mappings coded in SVG,
which transforms a rectangular repetitive pattern
into a rhombic pattern. The four transformations are linear.]]

In mathematics, a transformation, transform, or self-map{{Cite web|title=Self-Map -- from Wolfram MathWorld|url=https://mathworld.wolfram.com/Self-Map.html|access-date=March 4, 2024}} is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. {{nowrap|f: X → X}}.{{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|url=https://archive.org/details/classicalfinitet00gany_719|url-access=limited|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4|page=[https://archive.org/details/classicalfinitet00gany_719/page/n73 1]}}{{cite book|author=Pierre A. Grillet|title=Semigroups: An Introduction to the Structure Theory|url=https://books.google.com/books?id=yM544W1N2UUC&pg=PA2|year=1995|publisher=CRC Press|isbn=978-0-8247-9662-4|page=2}}{{cite book|author=Wilkinson, Leland |title=The Grammar of Graphics|publisher=Springer|year=2005|isbn=978-0-387-24544-7|page=29|url=https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA29|edition=2nd}}

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.{{Cite web|url=https://www.mathsisfun.com/geometry/transformations.html|title=Transformations|website=www.mathsisfun.com|access-date=2019-12-13}}{{Cite web|url=https://www.basic-mathematics.com/transformations-in-math.html|title=Types of Transformations in Math|website=Basic-mathematics.com|access-date=2019-12-13}}

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.{{cite book|author=Christopher Hollings|title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups|url=https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA251|year=2014|publisher=American Mathematical Society|isbn=978-1-4704-1493-1|page=251}}

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.{{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|url=https://archive.org/details/classicalfinitet00gany_719|url-access=limited|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4|page=[https://archive.org/details/classicalfinitet00gany_719/page/n74 2]}}

References

External links

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Category:Functions and mappings