Involution
From Wikipedia, the free encyclopedia
- See involution (philosophy) for the philosophical meaning.
In mathematics, an involution, or an involutary function, is a function that is its own inverse, so that
- f(f(x)) = x for all x in the domain of f.
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[edit] General properties
Any involution is a bijection.
The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation.
Other examples include circle inversion, the ROT13 transformation, and the Beaufort polyalphabetic cipher.
[edit] Involutions in Euclidean geometry
A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane. Doing a reflection twice, brings us back where we started.
This transformation is a particular case of an affine involution.
[edit] Involutions in differential geometry
In differential geometry, an involutive distribution is a certain type of subbundle of a vector bundle: the Lie bracket of any two vector fields in the sub-bundle stays in the sub-bundle. According to Frobenius' theorem, involutive distributions are completely integrable. They are often used to generate a foliation. Similarly, two functions f and g on a symplectic manifold are said to be in mutual involution if their Poisson bracket vanishes.
[edit] Involutions in ring theory
In ring theory, the word involution is customarily taken to mean an antihomomorphism that is its own inverse function. Examples include complex conjugation and the transpose of a matrix.
See also star-algebra.
[edit] Involutions in group theory
In group theory, an element of a group is an involution if it has order 2; i.e. an involution is an element a such that a2 = e, where e is the identity element. Originally, this definition differed not at all from the first definition above, since members of groups were always bijections from a set into itself, i.e., group was taken to mean permutation group. By the end of the 19th century, group was defined more broadly, and accordingly so was involution. The group of bijections generated by an involution through composition, is isomorphic with cyclic group C2.
A permutation is an involution precisely if it can be written as a product of non-overlapping transpositions.
The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the classification of finite simple groups.
Coxeter groups are groups generated by their involutions. Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions.
[edit] Involutions in mathematical logic
The operation of complement in Boolean algebras is an involution. Accordingly, negation in classical logic satisfies the law of double negation: ¬¬A is equivalent to A.
Generally in non-classical logics, negation which satisfies the law of double negation is called involutive. In algebraic semantics, such a negation is realized as an involution on the algebra of truth values. Examples of logics which have involutive negation are, e.g., Kleene and Bochvar three-valued logics, Łukasiewicz many-valued logic, fuzzy logic IMTL, etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual e.g. in formal fuzzy logic.
