View source for Involution (mathematics)

{{for|the archaic use of this term|exponentiation}}
[[Image:Involution.svg|right|thumb|An involution is a function <math>f:X\to X</math> that, when applied twice, brings one back to the starting point.]]
In [[mathematics]], an '''involution''', or an '''involutory function''', is a [[function (mathematics)|function]] {{mvar|f}} that is its own [[inverse function|inverse]],

: {{math|''f''(''f''(''x'')) {{=}} ''x''}}

for all {{mvar|x}} in the [[domain of a function|domain]] of {{mvar|f}}.<ref>{{Citation|last=Russell|first=Bertrand|title=Principles of mathematics|year=1903|publisher=W. W. Norton & Company, Inc|page=426|url=https://books.google.com/books?id=63ooitcP2osC&lpg=PR3&dq=involution%20subject%3A%|edition=2nd|isbn=9781440054167}}</ref> Equivalently, applying {{mvar|f}} twice produces the original value.

The term '''anti-involution''' refers to involutions based on [[antihomomorphism]]s (see ''{{section link|#Quaternion algebra, groups, semigroups}}'' below)

: {{math|''f''(''xy'') {{=}} ''f''(''y'') ''f''(''x'')}}

such that

: {{math|''xy'' {{=}} ''f''(''f''(''xy'')) {{=}} ''f''( ''f''(''y'') ''f''(''x'') ) {{=}} ''f''(''f''(''x''))  ''f''(''f''(''y'')) {{=}} ''xy''}}.

==General properties==
Any involution is a [[bijection]].

The [[identity function|identity map]] is a trivial example of an involution. Common examples in mathematics of nontrivial involutions include [[multiplication]] by &minus;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]], rotation by a half-turn, and [[reciprocal cipher]]s such as the [[ROT13]] transformation and the [[Beaufort cipher|Beaufort]] [[polyalphabetic cipher]].

The number of involutions, including the identity involution, on a set with {{nowrap|1=''n'' = 0, 1, 2, ...}} elements is given by a [[recurrence relation]] found by [[Heinrich August Rothe]] in 1800:
:''a''<sub>0</sub> = ''a''<sub>1</sub> = 1;
:''a''<sub>''n''</sub> = ''a''<sub>''n'' &minus; 1</sub> + (''n'' &minus; 1)''a''<sub>''n'' &minus; 2</sub>, for {{nowrap|''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}}; these numbers are called the [[Telephone number (mathematics)|telephone numbers]], and they also count the number of [[Young tableau]]x with a given number of cells.<ref>{{citation
 | last = Knuth | first = Donald E. | author-link = Donald Knuth
 | location = Reading, Mass.
 | mr = 0445948
 | pages = 48, 65
 | publisher = Addison-Wesley
 | title = [[The Art of Computer Programming]], Volume 3: Sorting and Searching
 | year = 1973}}.</ref>
The [[Function composition|composition]] {{nowrap|''g'' ∘ ''f''}} of two involutions ''f'' and ''g'' is an involution if and only if they commute: {{nowrap|1=''g'' ∘ ''f'' = ''f'' ∘ ''g''}}.<ref>{{citation|title=The Elements of Operator Theory|first=Carlos S.|last=Kubrusly|publisher=Springer Science & Business Media|year=2011|isbn=9780817649982|at=Problem&nbsp;1.11(a), p.&nbsp;27|url=https://books.google.com/books?id=g-UFYTO8SbMC&pg=PA27}}.</ref>

Every involution on an [[odd number]] of elements has at least one [[Fixed point (mathematics)|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 (mathematics)|parity]].<ref>{{citation
 | last = Zagier | first = D. | authorlink = Don Zagier
 | doi = 10.2307/2323918
 | issue = 2
 | journal = [[American Mathematical Monthly]]
 | mr = 1041893
 | page = 144
 | title = A one-sentence proof that every prime ''p''≡ 1 (mod 4) is a sum of two squares
 | volume = 97
 | year = 1990}}.</ref>

==Involution throughout the fields of mathematics==

===Pre-calculus ===
Basic examples of involutions are the functions:

:<math> f_1(x) = -x </math>, &nbsp; or &nbsp; <math> f_2(x) = \frac {1}{x} </math>, as well as their composition <math> (f_1\circ f_2)(x) = (f_2\circ f_1)(x) = f_3(x) = -\frac {1}{x}. </math> 

These are not the only pre-calculus involutions. Another one within the positive reals is:

:<math> f(x) = \ln\left(\frac {e^x+1}{e^x-1}\right) . </math>

The [[Graph_of_a_function|graph]] of an involution (on the real numbers) is [[Reflection_symmetry|line-symmetric]] over the line <math>y=x</math>. This is due to the fact that the inverse of any ''general'' function will be its reflection over the 45° line <math>y=x</math>. This can be seen by "swapping" <math>x</math> with <math>y</math>. If, in particular, the function is an ''involution'', then it will serve as its own reflection.

Other elementary involutions are useful in [[Functional equation#Solving functional equations|solving functional equations]].

===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. 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 (this is [[Desargues]]'s Involution Theorem,<ref>J. V. Field and J. J. Gray, ''The Geometrical Work of Girard Desargues,'' (New York: Springer, 1987), p. 54</ref> whose 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 (ed.), ''Selections Illustrating the History of Greek Mathematics,'' Volume II, number 362 in the [[Loeb Classical Library]] (Cambridge and London: Harvard and Heinemann, 1980), 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".

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

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

For a specific basis, any linear operator can be represented by a [[matrix (mathematics)|matrix]] ''T''. Every matrix has a [[transpose]], obtained by swapping rows for columns. This transposition is an involution on the set of matrices.

The definition of involution extends readily to [[module (mathematics)|modules]]. Given a module ''M'' over a [[ring (mathematics)|ring]] ''R'', an ''R'' [[endomorphism]] ''f'' of ''M'' is called an involution if ''f''&nbsp;<sup>2</sup> is the identity homomorphism on ''M''.

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

===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 {{nowrap|1=(''xy'')<sup>−1</sup> = ''y''<sup>−1</sup>''x''<sup>−1</sup>}}. 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]]
* multiplication by j 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; i.e. an involution is an element ''a'' such that ''a'' ≠ ''e'' and ''a''<sup>2</sup> = ''e'', where ''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; 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''.

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]].

An element ''x'' of a group ''G'' is called [[strongly real element|strongly real]] if there is an involution ''t'' {{nowrap begin}}with ''x''<sup>''t''</sup> = ''x''<sup>−1</sup>{{nowrap end}} {{nowrap begin}}(where ''x''<sup>''t''</sup> = ''t''<sup>−1</sup>⋅''x''⋅''t'').{{nowrap end}}

[[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]].

=== 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 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 value]]s. Examples of logics which have involutive negation are 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, 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-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).

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. XOR [[Mask (computing)|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. The [[Bitwise NOT|NOT]] bitwise operation is also an involution, and is a special case of the XOR operation where one parameter has all bits set to 1.

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 the form BGR.
f(f(RGB))=RGB, f(f(BGR))=BGR.

The [[RC4]] cryptographic cipher is an involution, as encryption and decryption operations use the same function.

== See also ==
* [[Automorphism]]
* [[Idempotence]]
* [[ROT13]]

== References ==
{{reflist}}

==Further reading==
* {{cite journal |title=Quaternion involutions and anti-involutions |first1=Todd A. |last1=Ell |first2=Stephen J. |last2=Sangwine |journal=Computers & Mathematics with Applications |volume=53 |issue=1 |year=2007 |pages=137–143 |doi=10.1016/j.camwa.2006.10.029|arxiv=math/0506034 }}
* {{citation | last1=Knus | first1=Max-Albert | last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Rost | first3=Markus | author3-link=Markus Rost | last4=Tignol | first4=Jean-Pierre | title=The book of involutions | others=With a preface by J. Tits | zbl=0955.16001 | series=Colloquium Publications | publisher=[[American Mathematical Society]] | volume=44 | location=Providence, RI | year=1998 | isbn=0-8218-0904-0 }}
* {{springer|title=Involution|id=p/i052510}}

[[Category:Abstract algebra]]
[[Category:Functions and mappings]]