Dictionary Definition
involution
Noun
1 reduction in size of an organ or part (as in
the return of the uterus to normal size after childbirth)
2 a long and intricate and complicated
grammatical construction
4 the act of sharing in the activities of a
group; "the teacher tried to increase his students' engagement in
class activities" [syn: engagement, participation, involvement] [ant: nonengagement,
nonengagement,
nonengagement]
5 the process of raising a quantity to some
assigned power [syn: exponentiation]
6 the action of enfolding something [syn:
enfolding]
User Contributed Dictionary
English
Noun
 entanglement; a
spiralling inwards; intricacy

 1968: ‘Gomez,’ said the mortician, ‘is an expert only on the involutions of his own rectum.’ — Anthony Burgess, Enderby Outside

 An endofunction whose square is equal to the identity function; a function equal to its inverse.
 The regressive changes in the body occurring with old age.
Derived terms
Translations
 Italian: involuzione
Extensive Definition
In mathematics, an involution,
or an involutary function, is a function
that is its own inverse,
so that
 f(f(x)) = x for all x in the domain of f.
General properties
Any involution is a bijection.
The 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
reciprocals,
complementation
in set
theory and complex
conjugation.
Other examples include circle
inversion, the ROT13 transformation,
and the Beaufort
polyalphabetic
cipher.
Involutions in Euclidean geometry
A simple example of an involution of the
threedimensional Euclidean
space is reflection
against a 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 T^2=I. 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.
Involutions are related to idempotents; 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 staralgebra.
Involutions in 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 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 C2.
A permutation is an involution
precisely if it can be written as a product of one or more
nonoverlapping 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.
Involutions in 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 nonclassical 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,
e.g., Kleene and Bochvar threevalued
logics, Łukasiewicz
manyvalued logic, fuzzy logic
IMTL, etc. Involutive negation is sometimes added as an additional
connective to logics with noninvolutive negation; this is usual
e.g. in tnorm
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 MValgebras and
BLalgebras (and
so correspondingly between Łukasiewicz
logic and fuzzy logic BL), IMTL and
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.
See also
involution in German: Involution
(Mathematik)
involution in Esperanto: Involucio
involution in French: Involution
(mathématiques)
involution in Italian: Involuzione (teoria degli
insiemi)
involution in Dutch: Involutie (wiskunde)
involution in Japanese: 対合
involution in Polish: Inwolucja
(matematyka)
involution in Portuguese: Involução
(matemática)
involution in Russian: Инволюция
(математика)
involution in Slovak: Involúcia
(matematika)
involution in Serbian: Инволуција
(математика)
involution in Chinese: 對合
Synonyms, Antonyms and Related Words
absorption, addition, ambages, anfractuosity, approximation, circuitousness, circumambages, circumbendibus, circumlocution, circumvolution, comedown, complexity, complexness, complication, convolution, crabbedness, crinkle, crinkling, debasement, decadence, decadency, declension, declination, decline, deformation, degeneracy, degenerateness, degeneration, degradation, demotion, depravation, depravedness, depreciation, derogation, descent, deterioration, devolution, differentiation,
division, downtrend, downturn, downward mobility,
downward trend, drop,
dying, ebb, effeteness, embarrassment, engagement, enmeshment, entanglement, equation, evolution, extrapolation, fading, failing, failure, failure of nerve,
fall, fallingoff, flexuosity, flexuousness, implication, inclusion, integration, interpolation, intorsion, intricacy, intricateness, inversion, involvement, lapse, loss of tone, meander, meandering, multiplication, notation, perplexity, practice, proportion, ramification, reduction, regression, relation, retrocession, retrogradation, retrogression, rivulation, sinuation, sinuosity, sinuousness, slinkiness, slippage, slump, snakiness, subtlety, subtraction, tanglement, technicality, torsion, tortility, tortuosity, tortuousness, transformation, turning, twisting, undulation, wane, wave, waving, winding