Involution (mathematics)

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".

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.

Involutions in Euclidean geometry

A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane. Performing a reflection twice brings us back where we started.

This transformation is a particular case of an affine involution.

Involutions in linear algebra

In linear algebra, an involution is a linear operator "T" such that $T^2=I$. Except for in characteristic 2, such operators are diagonalizable with 1's and -1's on the diagonal. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable.

Involutions are related to idempotents; if 2 is invertible, (in a field of characteristic other than 2), then they are equivalent.

Involutions in ring theory

In ring theory, the word "involution" is customarily taken to mean an antihomomorphism that is its own inverse function.Examples of involutions in common rings:
* complex conjugation on the complex plane
* multiplication by j in the split-complex numbers
* taking the transpose in a matrix ring

"See also star-algebra."

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 "a" ≠ "e" and "a"² = "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 "C"₂.

A permutation is an involution precisely if it can be written as a product of one or more 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 involutions with the relations determined only by relations given for pairs of the generating involutions. Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions.

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 t-norm fuzzy logics.

The involutiveness of negation is an important characterization property for logics and the corresponding varieties of algebras. For instance, involutive negation characterizes Boolean algebras among Heyting algebras. Correspondingly, classical Boolean logic arises by adding the law of double negation to intuitionistic logic. The same relationship holds also between MV-algebras and BL-algebras (and so correspondingly between Łukasiewicz logic and fuzzy logic BL), IMTL and MTL, and other pairs of important varieties of algebras (resp. corresponding logics).

Count of involutions

The number of involutions on a set with "n" = 0, 1, 2, ... elements is given by the recurrence relation:

:"a"(0) = "a"(1) = 1;:"a"("n") = "a"("n" − 1) + ("n" − 1) × "a"("n" − 2), for "n" > 1.The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 OEIS|id=A000085.

See also

* Automorphism