Editing Involution (mathematics) (section)

== Involution throughout the fields of mathematics ==

=== Real-valued functions ===

The [[Graph_of_a_function|graph]] of an involution (on the real numbers) is [[Reflection_symmetry|symmetric]] across the line {{math|1=''y'' = ''x''}}. This is due to the fact that the inverse of any ''general'' function will be its reflection over the line {{math|1=''y'' = ''x''}}. This can be seen by "swapping" {{mvar|x}} with {{mvar|y}}. If, in particular, the function is an ''involution'', then its graph is its own reflection. 

Some basic examples of involutions include the functions
<math display="block">\begin{alignat}{1}
f(x) &= a-x \; , \\
f(x) &= \frac{b}{x-a}+a
\end{alignat}</math>Besides, we can construct an involution by wrapping an involution {{mvar|g}} in a bijection {{mvar|h}} and its inverse (<math>h^{-1} \circ  g \circ h</math>). For instance :<math display="block">\begin{alignat}{2}
f(x) &= \sqrt{1 - x^2} \quad\textrm{on}\; [0;1] & \bigl(g(x) = 1-x \quad\textrm{and}\quad h(x) = x^2\bigr), \\
f(x) &= \ln\left(\frac {e^x+1}{e^x-1}\right) & \bigl(g(x) = \frac{x+1}{x-1}=\frac{2}{x-1}+1 \quad\textrm{and}\quad h(x) = e^x\bigr)  \\
\end{alignat}</math>
=== Euclidean geometry ===
A simple example of an involution of the three-dimensional [[Euclidean space]] is [[Reflection (mathematics)|reflection]] through a [[plane (mathematics)|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 involution]]s.

=== Projective geometry ===
An involution is a [[projectivity]]  of period 2, that is, a projectivity that interchanges pairs of points.<ref name=AGP>A.G. Pickford (1909) [https://archive.org/details/elementaryprojec00pickrich/page/n5 Elementary Projective Geometry], [[Cambridge University Press]] via [[Internet Archive]]</ref>{{rp|24}}
* 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. This theorem has been called [[Desargues]]'s Involution Theorem.<ref>[[Judith V. Field|J. V. Field]] and J. J. Gray (1987) ''The Geometrical Work of Girard Desargues'', (New York: Springer), p. 54</ref> Its origins can be seen in Lemma IV of the lemmas to the ''Porisms'' of Euclid in Volume VII of the ''Collection'' of [[Pappus of Alexandria]].<ref>Ivor Thomas (editor) (1980) ''Selections Illustrating the History of Greek Mathematics'', Volume II, number 362 in the [[Loeb Classical Library]] (Cambridge and London: Harvard and Heinemann), pp. 610&ndash;3</ref>
* If an involution has one [[fixed point (mathematics)|fixed point]], it has another, and consists of the correspondence between [[projective harmonic conjugate|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 the context of projectivities, fixed points are called '''double points'''.<ref name=AGP/>{{rp|53}}

Another type of involution occurring in projective geometry is a '''polarity''' that is a [[correlation (projective geometry)|correlation]] of period 2.<ref>[[H. S. M. Coxeter]] (1969) ''Introduction to Geometry'', pp. 244–8, [[John Wiley & Sons]]</ref>

=== Linear algebra ===
{{details|Involutory matrix}}
In linear algebra, an involution is a linear operator {{math|''T''}} on a vector space, such that {{math|1={{itco|''T''}}{{sup|2}} = ''I''}}. Except for in characteristic 2, such operators are diagonalizable for a given basis with just {{math|1}}s and {{math|&minus;1}}s 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 {{math|''V''}} is chosen, and that {{math|''e''<sub>1</sub>}} and {{math|''e''<sub>2</sub>}} are basis elements. There exists a linear transformation {{math|''f''}} that sends {{math|''e''<sub>1</sub>}} to {{math|''e''<sub>2</sub>}}, and sends {{math|''e''<sub>2</sub>}} to {{math|''e''<sub>1</sub>}}, and that is the identity on all other basis vectors. It can be checked that {{math|1=''f''(''f''(''x'')) = ''x''}} for all {{math|''x''}} in {{math|''V''}}. That is, {{math|''f''}} is an involution of {{math|''V''}}.

For a specific basis, any linear operator can be represented by a [[matrix (mathematics)|matrix]] {{math|''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 [[module (mathematics)|modules]]. Given a module {{math|''M''}} over a [[ring (mathematics)|ring]] {{math|''R''}}, an {{math|''R''}} [[endomorphism]] {{math|''f''}} of {{math|''M''}} is called an involution if {{math|{{itco|''f''}}<sup>2</sup>}} is the identity homomorphism on {{math|''M''}}.

[[Idempotent element (ring theory)#Relation with involutions|Involutions are related to idempotent]]s; if {{math|2}} is invertible then they [[bijection|correspond]] in a one-to-one manner.

In [[functional analysis]], [[Banach *-algebra]]s and [[C*-algebra]]s are special types of [[Banach algebra]]s with involutions.

=== Quaternion algebra, groups, semigroups ===
In a [[quaternion algebra]], an (anti-)involution is defined by the following axioms: if we consider a transformation <math>x \mapsto f(x)</math> then it is an involution if
* <math> f(f(x))=x </math> (it is its own inverse)
* <math> f(x_1+x_2)=f(x_1)+f(x_2) </math> and <math> f(\lambda x)=\lambda f(x) </math> (it is linear)
* <math> f(x_1 x_2)=f(x_1) f(x_2) </math>

An anti-involution does not obey the last axiom but instead
* <math> f(x_1 x_2)=f(x_2) f(x_1) </math>

This former law is sometimes called [[antidistributive]]. It also appears in [[group (mathematics)|groups]] as {{math|1=(''xy''){{sup|−1}} = (''y''){{sup|−1}}(''x''){{sup|−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.

=== Ring theory ===
{{details|*-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:
* [[complex conjugation]] on the [[complex plane]], and its equivalent in the [[split-complex number]]s
* taking the transpose in a matrix ring.

=== Group theory ===
In [[group theory]], an element of a [[group (mathematics)|group]] is an involution if it has [[order (group theory)|order]] 2; that is, an involution is an element {{math|''a''}} such that {{math|''a'' ≠ ''e''}} and {{math|1=''a''<sup>2</sup> = ''e''}}, where {{math|''e''}} is the [[identity element]].<ref>
John S. Rose.
[https://books.google.com/books?id=j-I7Zpq3GdIC "A Course on Group Theory"].
p. 10, section 1.13.
</ref>
Originally, this definition agreed with the first definition above, since members of groups were always bijections from a set into itself; that is, ''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 if and only if it can be written as a finite product of disjoint [[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]].

An element {{math|''x''}} of a group {{math|''G''}} is called [[strongly real element|strongly real]] if there is an involution {{math|''t''}} with&nbsp;{{math|1=''x''{{sup|''t''}} = ''x''{{sup|−1}}}} (where&nbsp;{{math|1=''x''{{sup|''t''}} = ''x''{{sup|−1}} = ''t''{{sup|−1}} ⋅ ''x'' ⋅ ''t''}}).

[[Coxeter group]]s are groups generated by a set {{math|''S''}} of involutions subject only to relations involving powers of pairs of elements of {{math|''S''}}.  Coxeter groups can be used, among other things, to describe the possible [[Platonic solid|regular polyhedra]] and their [[regular polytope|generalizations to higher dimensions]].

=== Mathematical logic ===
The operation of complement in [[Boolean algebra (structure)|Boolean algebra]]s is an involution. Accordingly, [[negation]] in [[classical logic]] satisfies the ''[[Double_negation|law of double negation]]'': {{math|¬¬''A''}} is equivalent to {{math|''A''}}.

Generally in [[non-classical logic]]s, negation that satisfies the law of double negation is called ''involutive''. In [[Algebraic semantics (mathematical logic)|algebraic semantics]], such a negation is realized as an involution on the algebra of [[truth value]]s. Examples of logics that have involutive negation are Kleene and Bochvar [[three-valued logic]]s, [[Łukasiewicz logic|Łukasiewicz many-valued logic]], the [[fuzzy logic]] '[[monoidal t-norm logic|involutive monoidal t-norm 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 [[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 (logic)|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 (respectively, corresponding logics).

In the study of [[binary relation]]s, 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 [[partial order|ordered]] through [[inclusion (set theory)|inclusion]]. While this ordering is reversed with the [[complementation (mathematics)|complementation]] involution, it is preserved under conversion.

=== Computer science ===
The [[XOR]] [[bitwise operation]] with a given value for one parameter is an involution on the other parameter. XOR [[Mask (computing)|masks]] in some instances were used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state.

Two special cases of this, which are also involutions, are the [[bitwise NOT]] operation which is XOR with an all-ones value, and [[stream cipher]] [[encryption]], which is an XOR with a secret [[keystream]].

This predates binary computers; 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.<ref>{{cite book
 |title=Classical Cryptology
 |first=Greg |last=Goebel
 |chapter-url=http://vc.airvectors.net/ttcode_05.html
 |chapter=The Mechanization of Ciphers
 |year=2018
}}</ref>

Another involution used in computers is an order-2 bitwise permutation.  For example. a color value stored as integers in the form {{math|(''R'', ''G'', ''B'')}}, could exchange {{math|''R''}} and {{math|''B''}}, resulting in the form {{math|(''B'', ''G'', ''R'')}}: {{math|1=''f''(''f''(RGB)) = RGB, ''f''(''f''(BGR)) = BGR}}.
=== Physics ===
[[Legendre transformation]], which converts between the [[Lagrangian mechanics|Lagrangian]] and [[Hamiltonian mechanics|Hamiltonian]], is an involutive operation. 

[[Integrability (disambiguation)|Integrability]], a central notion of physics and in particular the subfield of [[Integrable system|integrable systems]], is closely related to involution, for example in context of [[Kramers–Wannier duality]].