Involution

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Involution

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an involution, or an involutary function, is a function

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
that is its own inverse

Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
, so that

f(f(x)) = x for all x in the domain

Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
of f.

Involutions in Euclidean geometry
A simple example of an involution of the three-dimensional Euclidean space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
is reflection

Reflection (mathematics)

In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
against a plane

Plane (mathematics)

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In mathematics

Mathematics

Function (mathematics)

Inverse function

f(f(x)) = x for all x in the domain

Domain (mathematics)

General properties

Any involution is a bijection

Bijection

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y....
.

The identity map

Identity function

In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument....
is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication

Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
by −1 in arithmetic

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations....
, the taking of reciprocals

Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1⁄x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1....
, complementation

Complement (set theory)

In discrete mathematics and predominantly in set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation to another....
in set theory

Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
and complex conjugation

Complex conjugate

In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number...
.

Other examples include circle inversion, the ROT13

ROT13

ROT13 is a simple substitution cipher used in online forums as a means of hiding spoiler s, punch line, puzzle solutions, and profanitys from the casual glance....
transformation, and the Beaufort

Beaufort cipher

The Beaufort cipher, created by Sir Francis Beaufort, is a substitution cipher that is similar to the Vigen?re cipher but uses a slightly modified enciphering mechanism and cryptographic tableau....
polyalphabetic cipher

Polyalphabetic cipher

A polyalphabetic cipher is any cipher based on substitution cipher, using multiple substitution alphabets. The Vigen?re cipher is probably the best-known example of a polyalphabetic cipher, though it is a simplified special case....
.

Involutions in Euclidean geometry

A simple example of an involution of the three-dimensional Euclidean space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
is reflection

Reflection (mathematics)

Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
. Performing a reflection twice brings us back where we started.

Another is the so-called reflection through the origin

Reflection through the origin

In mathematics, reflection through the origin refers to the orthogonal transform of , also written or scalar multiplication by . In coordinates, in two dimensions, this is the map that sends , in three dimensions, this sends , and so forth....
; this is an abuse of language, as it is an involution, but not a reflection

Reflection (mathematics)

Affine involution

In Euclidean geometry, of special interest are involutions which are linear transformation or affine transformations over the Euclidean space Rn....
s.

Involutions in linear algebra

In linear algebra, an involution is a linear operator T such that . 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.

The transpose of a square matrix is an involution because the transpose of the transpose of a matrix is itself.

Involutions are related to idempotent

Idempotence

Idempotence describes the property of operations in mathematics and computer science which means that multiple applications of the operation does not change the result....
s; if 2 is invertible, (in a field of characteristic other than 2), then they are equivalent.

Involutions in ring theory

In ring theory

Ring theory

In mathematics, ring theory is the study of ring , algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers....
, the word involution is customarily taken to mean an antihomomorphism

Antihomomorphism

In mathematics, an antihomomorphism is a type of Function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism which has an inverse as an antihomomorphism; this coincides with it being a bijection from an object to itself....
that is its own inverse function. Examples of involutions in common rings:

complex conjugation on the complex plane
Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
multiplication by j in the split-complex number
Split-complex number

In linear algebra, a split-complex number is of the form z = x +y j where j2 = +1 , and x and y are real numbers....
s
taking the transpose
Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
in a matrix ring

See also star-algebra
Star-algebra

A *-operation on a *-ring is an operation on a ring that behaves similarly to complex conjugation on the complex numbers. A *-operation on a *-algebra is an operation on an algebra over a *-ring that behaves similarly to taking conjugate transpose in ....
.

Involutions in group theory

In group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
, an element of a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
is an involution if it has order

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two closely related senses:* the order of a group is its cardinality, i.e....
2; i.e. an involution is an element a such that a ? e and a² = e, where e is the identity element

Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
. 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
Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given Set M, and whose group operation is the composition of permutations in G ; the relationship is often written as ....
. 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

Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
C₂.

A permutation

Permutation

In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
is an involution precisely if it can be written as a product of one or more non-overlapping transposition

Transposition (mathematics)

In informal language, a transposition is a function that swaps two elements of a set. More formally, given a finite set Set , a transposition is a permutation such that there exist indices such that , and for all other indices This is often denoted as ...
s.

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

Classification of finite simple groups

The classification of the finite simple groups, also called the enormous theorem, is believed to classify all List of finite simple groups. These group can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic building blocks of the natural numbers....
.

Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
s 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

Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
and their generalizations to higher dimensions

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n....
.

Involutions in mathematical logic

The operation of complement in Boolean algebras is an involution. Accordingly, negation

Negation

In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true....
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 value

Logical value

In logic and mathematics, a logical value, also called a truth value, is a value indicating the extent to which a proposition is truth.In classical logic, the only possible truth values are true and false....
s. Examples of logics which have involutive negation are, e.g., Kleene and Bochvar three-valued logics, Lukasiewicz many-valued logic

Lukasiewicz logic

In mathematics, Lukasiewicz logic is a non-classical logic, many-valued logic logic. It was originally defined by Jan Lukasiewicz as a three-valued logic; it was later generalized to n-valued as well as infinitely-many-valued variants, both propositional and first-order....
, fuzzy logic

Fuzzy logic

Fuzzy logic is a form of multi-valued logic derived from fuzzy set theory to deal with reasoning that is approximate rather than precise. In binary sets with binary logic, in contrast to fuzzy logic named also crisp logic, the variables may have a Membership function of only 0 or 1....
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

T-norm fuzzy logics

T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics which takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of logical conjunction....
.

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

Variety (universal algebra)

In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of mathematical identity....
. For instance, involutive negation characterizes Boolean algebras among Heyting algebra

Heyting algebra

In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebra s, named after Arend Heyting....
s. Correspondingly, classical Boolean logic

Classical logic

Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. They are characterised by a number of properties; non-classical logics are those that lack one or more of these properties, which are:...
arises by adding the law of double negation to intuitionistic logic

Intuitionistic logic

Intuitionistic logic, or constructivist logic, is the symbolic logic system originally developed by Arend Heyting to provide a formal basis for Luitzen Egbertus Jan Brouwer's programme of intuitionism....
. The same relationship holds also between MV-algebra

MV-algebra

In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation , a unary operation , and the constant , satisfying certain axioms....
s and BL-algebras (and so correspondingly between Lukasiewicz logic

Lukasiewicz logic

BL (logic)

Basic fuzzy Logic , the logic of continuous function t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of all left-continuous t-norms Monoidal t-norm logic....
), IMTL and MTL

Monoidal t-norm logic

Monoidal t-norm based logic , the logic of left-continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of commutative bounded integral residuated lattices by the axiom of prelinearity....
, 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

Recurrence relation

In mathematics, a recurrence relation is an equation that defines a sequence recursion: each term of the sequence is defined as a Function of the preceding terms....
:

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 (number)

1 is a number, number names, and the name of the glyph representing that number.It represents a single entity, the unit of counting or measurement....
, 1, 2

2 (number)

2 is a number, numeral, and glyph. It is the natural number following 1 and preceding 3 ....
, 4

4 (number)

This article discusses the number Four. For the year 4 AD, see 4. For other uses of 4, see 4 4 is a number, numeral, and glyph....
, 10

10 (number)

10 is an Even and odd numbers natural number following 9 and preceding 11 ....
, 26

26 (number)

26 is the natural number following 25 and preceding 27 ....
, 76

76 (number)

76 is the natural number following 75 and preceding 77 ....
, 232 .