Involution (mathematics)

Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher.

Thus the number of fixed points of all the involutions on a given finite set have the same parity.

In particular, every involution on an odd number of elements has at least one fixed point.

[5] The graph of an involution (on the real numbers) is symmetric across the line y = x.

This is due to the fact that the inverse of any general function will be its reflection over the line y = x.

A simple example of an involution of the three-dimensional Euclidean space is reflection through a plane.

Performing a reflection twice brings a point back to its original coordinates.

[6]: 24 Another type of involution occurring in projective geometry is a polarity that is a correlation of period 2.

Except for in characteristic 2, such operators are diagonalizable for a given basis with just 1s and −1s on the diagonal of the corresponding matrix.

It can be checked that f(f(x)) = x for all x in V. That is, f is an involution of V. For a specific basis, any linear operator can be represented by a matrix T. Every matrix has a transpose, obtained by swapping rows for columns.

Given a module M over a ring R, an R endomorphism f of M is called an involution if f2 is the identity homomorphism on M. Involutions are related to idempotents; if 2 is invertible then they correspond in a one-to-one manner.

In a quaternion algebra, an (anti-)involution is defined by the following axioms: if we consider a transformation

then it is an involution if An anti-involution does not obey the last axiom but instead This former law is sometimes called antidistributive.

Taken as an axiom, it leads to the notion of semigroup with involution, of which there are natural examples that are not groups, for example square matrix multiplication (i.e. the full linear monoid) with transpose as the involution.

In ring theory, the word involution is customarily taken to mean an antihomomorphism that is its own inverse function.

By the end of the 19th century, group was defined more broadly, and accordingly so was involution.

A permutation is an involution if and only if it can be written as a finite product of disjoint transpositions.

The study of involutions was instrumental in the classification of finite simple groups.

Coxeter groups are groups generated by a set S of involutions subject only to relations involving powers of pairs of elements of S. Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions.

Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual, for example, in t-norm fuzzy logics.

The involutiveness of negation is an important characterization property for logics and the corresponding varieties of algebras.

While this ordering is reversed with the complementation involution, it is preserved under conversion.

XOR masks in some instances were used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state.

Two special cases of this, which are also involutions, are the bitwise NOT operation which is XOR with an all-ones value, and stream cipher encryption, which is an XOR with a secret keystream.

This predates binary computers; practically all mechanical cipher machines implement a reciprocal cipher, an involution on each typed-in letter.

a color value stored as integers in the form (R, G, B), could exchange R and B, resulting in the form (B, G, R): f(f(RGB)) = RGB, f(f(BGR)) = BGR.

Legendre transformation, which converts between the Lagrangian and Hamiltonian, is an involutive operation.

Integrability, a central notion of physics and in particular the subfield of integrable systems, is closely related to involution, for example in context of Kramers–Wannier duality.