In mathematics, an involution, involutory function, or self-inverse function is a function f that is its own inverse,
for all x in the domain of f. Equivalently, applying f twice produces the original value.
Any involution is a bijection.
The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () 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.
The composition g ∘ f of two involutions f and g is an involution if and only if they commute: g ∘ f = f ∘ g.
The number of involutions, including the identity involution, on a set with n = 0, 1, 2, ... elements is given by a recurrence relation found by Heinrich August Rothe in 1800:
The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (sequence A000085 in the OEIS); these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells. The number can also be expressed by non-recursive formulas, such as the sum
The number of fixed points of an involution on a finite set and its number of elements have the same parity. 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. This can be used to prove Fermat's two squares theorem.
Some basic examples of involutions include the functions
These are not the only pre-calculus involutions. Another one within the positive reals is
The graph of an involution (on the real numbers) is symmetric across the line . This is due to the fact that the inverse of any general function will be its reflection over the line . This can be seen by "swapping" with . If, in particular, the function is an involution, then its graph is its own reflection.
Other elementary involutions are useful in solving functional equations.
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.
Another involution is reflection through the origin; not a reflection in the above sense, and so, a distinct example.
These transformations are examples of affine involutions.
An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points.: 24
Another type of involution occurring in projective geometry is a polarity which is a correlation of period 2. 
Further information: Involutory matrix
In linear algebra, an involution is a linear operator T on a vector space, such that . 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. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable.
For example, suppose that a basis for a vector space V is chosen, and that e1 and e2 are basis elements. There exists a linear transformation f which sends e1 to e2, and sends e2 to e1, and which is the identity on all other basis vectors. 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. This transposition is an involution on the set of matrices. Since elementwise complex conjugation is an independent involution, the conjugate transpose or Hermitian adjoint is also an involution.
The definition of involution extends readily to modules. Given a module M over a ring R, an R endomorphism f of M is called an involution if f 2 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 functional analysis, Banach *-algebras and C*-algebras are special types of Banach algebras with involutions.
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. It also appears in groups as . 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.
Further information: *-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:
In group theory, an element of a group is an involution if it has order 2; i.e. an involution is an element such that 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 and only if it can be written as a finite product of disjoint 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.
An element of a group is called strongly real if there is an involution with (where ).
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.
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 that 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).
In the study of binary relations, every relation has a converse relation. Since the converse of the converse is the original relation, the conversion operation is an involution on the category of relations. Binary relations are ordered through inclusion. While this ordering is reversed with the complementation involution, it is preserved under conversion.
The XOR bitwise operation with a given value for one parameter is 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. The NOT bitwise operation is also an involution, and is a special case of the XOR operation where one parameter has all bits set to 1.
Another example is a bit mask and shift function operating on color values stored as integers, say in the form RGB, that swaps R and B, resulting in the form BGR. f(f(RGB))=RGB, f(f(BGR))=BGR.
The RC4 cryptographic cipher is an involution, as encryption and decryption operations use the same function.
Practically all mechanical cipher machines implement a reciprocal cipher, an involution on each typed-in letter. Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way.