f(f(x)) = x
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.
- 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); these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells. The composition of two involutions f and g is an involution if and only if they commute: .
Every involution on an odd number of elements has at least one fixed point. More generally, for an involution on a finite set of elements, the number of elements and the number of fixed points have the same parity.
Involution throughout the fields of mathematics
These transformations are examples of affine involutions.
An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points. Coxeter relates three theorems on involutions:
- Any projectivity that interchanges two points is an involution.
- The three pairs of opposite sides of a complete quadrangle meet any line (not through a vertex) in three pairs of an involution.
- If an involution has one fixed point, it has another, and consists of the correspondence between harmonic conjugates with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic".
In linear algebra, an involution is a linear operator T such that . 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.
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.
Quaternion algebra, groups, semigroups
In a quaternion algebra, an (anti-)involution is defined by the following axioms: if we consider a transformation then an involution is
- . An involution is its own inverse
- An involution is linear: and
An anti-involution does not obey the last axiom but instead
This former law is sometimes called antidistributive. It also appears in groups as (xy)-1 = y-1x-1. 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.
- complex conjugation on the complex plane
- multiplication by j in the split-complex numbers
- taking the transpose in a matrix ring.
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.
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.
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).
The XOR bitwise operation with a given value for one parameter 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.
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 form BGR. f(f(RGB))=RGB, f(f(BGR))=BGR.
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