Involution
293450
199401097
2008-03-19T18:43:52Z
Lambiam
745100
/* Involutions in group theory */ e has order 1
{{otheruses4|involution in mathematics|other uses}} <!--involution of the uterus|Involution (biology)}}-->
[[Image:Involution.svg|right|thumb|An involution is a function <math>f:X\to X,</math> which, when applied twice, brings one back to the starting point.]]
In [[mathematics]], an '''involution''', or an '''involutary function''', is a [[function (mathematics)|function]] that is its own [[inverse function|inverse]], so that
:''f''(''f''(''x'')) = ''x'' for all ''x'' in the [[domain (mathematics)|domain]] of ''f''.
==General properties==
Any involution is a [[bijection]].
The [[identity function|identity map]] is a trivial example of an involution. Common examples in mathematics of more interesting involutions include [[multiplication]] by −1 in [[arithmetic]], the taking of [[multiplicative inverse|reciprocals]], [[complement (set theory)|complementation]] in [[set theory]] and [[complex conjugate|complex conjugation]].
Other examples include [[circle inversion]], the [[ROT13]] transformation, and the [[Beaufort cipher|Beaufort]] [[polyalphabetic cipher]].
==Involutions in Euclidean geometry==
A simple example of an involution of the three-dimensional [[Euclidean space]] is [[Reflection (mathematics)|reflection]] against a [[plane (mathematics)|plane]]. Doing a reflection twice, brings us back where we started.
This transformation is a particular case of an [[affine involution]].
==Involutions in linear algebra==
In linear algebra, an involution is a linear operator ''T'' such that <math>T^2=I</math>. 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.
[[Idempotence#Relation with involutions|Involutions are related to idempotent]]s; if 2 is invertible, (in a field of characteristic other than 2), then they are equivalent.
==Involutions in ring theory==
In [[ring theory]], the word ''involution'' is customarily taken to mean an [[antihomomorphism]] that is its own inverse function. Examples include [[complex conjugation]] and the [[transpose]] of a matrix.
''See also [[star-algebra]].''
==Involutions in group theory==
In [[group theory]], an element of a [[group (mathematics)|group]] is an involution if it has [[order (group theory)|order]] 2; i.e. an involution is an element ''a'' such that ''a'' ≠ ''e'' and ''a''<sup>2</sup> = ''e'', where ''e'' is the [[identity element]]. 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]]''. 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]] ''C''<sub>2</sub>.
A [[permutation]] is an involution precisely if it can be written as a product of one or more non-overlapping [[transposition (mathematics)|transposition]]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]].
[[Coxeter group]]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 [[Platonic solid|regular polyhedra]] and their [[regular polytope|generalizations to higher dimensions]].
== Involutions in mathematical logic ==
The operation of complement in [[Boolean algebra (structure)|Boolean algebra]]s 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 [[logical value|truth value]]s. Examples of logics which have involutive negation are, e.g., Kleene and Bochvar [[three-valued logic]]s, [[Łukasiewicz logic|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 e.g. in [[t-norm fuzzy logics]].
The involutiveness of negation is an important characterization property for logics and the corresponding [[variety (universal algebra)|varieties of algebras]]. For instance, involutive negation characterizes [[Boolean algebra (structure)|Boolean algebra]]s among [[Heyting algebra]]s. Correspondingly, classical [[classical logic|Boolean logic]] arises by adding the law of double negation to [[intuitionistic logic]]. The same relationship holds also between [[MV-algebra]]s and [[BL-algebra]]s (and so correspondingly between [[Łukasiewicz logic]] and fuzzy logic [[BL (logic)|BL]]), IMTL and [[Monoidal t-norm logic|MTL]], 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]]:
:''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 (number)|1]], 1, [[2 (number)|2]], [[4 (number)|4]], [[10 (number)|10]], [[26 (number)|26]], [[76 (number)|76]], [[232 (number)|232]] {{OEIS|id=A000085}}.
== See also ==
* [[Automorphism]]
[[Category:Abstract algebra]]
[[Category:Functions and mappings]]
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