Involution (mathematics)

In mathematics, an (anti-)involution, or an involutary function, is a function f that is its own inverse:

f(f(x)) = x for all x in the domain of f. [1]

Contents

General properties

Any involution is a bijection.

The identity map is a trivial example of an involution. Common examples in mathematics of more detailed involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other examples include circle inversion, rotation by a half-turn, and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher.

The number of involutions, including the identity involution, on a set with n = 0, 1, 2, … elements is given by the recurrence relation:

a0 = a1 = 1;
an = an − 1 + (n − 1)an − 2, for n > 1.

The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (sequence A000085 in OEIS).

Involution throughout the fields of mathematics

Euclidean geometry

A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane. Performing a reflection twice brings a point back to its original coordinates.

Another is the so-called reflection through the origin; this is an abuse of language, as it is actually an involution, not a reflection.

These transformations are examples of affine involutions.

Projective geometry

An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points. Coxeter relates three theorems on involutions:

Another type of involution occurring in projective geometry is a polarity which is a correlation of period 2. [2]

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 1s and −1s on the diagonal. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable.

The transpose of a square matrix is an involution; because the transpose, of the transpose of a matrix, is the original matrix.

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

Quaternion algebra

In a Quaternion algebra, an (anti-)involution is defined by the following axioms: if we consider a transformation \begin{align} x &\mapsto f(x) \end{align} then an involution is

An anti-involution does not obey the last axiom but instead

Ring theory

For more details on this topic, see *-algebra.

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:

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 a2 = e, where e is the identity element. Originally, this definition agreed with 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.

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.

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 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, for example, 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).

Computer science

The XOR bitwise operation is also an involution. XOR masks were once used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state.

References

  1. ^ Russell, Bertrand (1903). Principles of mathematics (2nd Edition ed.). W. W. Norton & Company, Inc. pp. page 426. http://books.google.com/books?id=63ooitcP2osC&lpg=PR3&dq=involution%20subject%3A%. 
  2. ^ H. S. M. Coxeter (1969) Introduction to Geometry, pp 244–8, John Wiley & Sons

Todd A. Ell; Stephen J. Sangwine (2007), "Quaternion involutions and anti-involutions", Computers & Mathematics with Applications 53 (1): 137–143, doi:10.1016/j.camwa.2006.10.029 

See also