# Involution (mathematics)

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{{Short description|Function that is its own inverse}}
{{Other uses|Involution (disambiguation)#Mathematics}}
right|thumb|An involution is a function {{math|''f'' : ''X'' → ''X''}} that, when applied twice, brings one back to the starting point.

In [mathematics](/source/mathematics), an '''involution''', '''involutory function''', or '''self-inverse function'''<ref>Robert Alexander Adams, ''Calculus: Single Variable'', 2006, {{isbn|0321307143}}, p. 165</ref> is a [function](/source/function_(mathematics)) {{mvar|f}} that is its own [inverse](/source/inverse_function), {{math|1=''f''(''f''(''x'')) = ''x''}}
for all {{mvar|x}} in the [domain](/source/domain_of_a_function) of {{math|''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&q=involution%20subject%3A%&pg=PR3|edition=2nd|isbn=9781440054167}}</ref> Equivalently, applying {{mvar|f}} twice produces the original value.

== General properties ==
Any involution is a [bijection](/source/bijection).

The [identity map](/source/identity_function) is a trivial example of an involution. Examples of nontrivial involutions include [negation](/source/Additive_inverse) ({{math|''x'' ↦ −''x''}}), [reciprocation](/source/multiplicative_inverse) ({{math|''x'' ↦ 1/''x''}}), and [complex conjugation](/source/complex_conjugate) ({{math|''z'' ↦ {{overline|''z''}}}}) in [arithmetic](/source/arithmetic); [reflection](/source/reflection_(mathematics)), half-turn [rotation](/source/rotation_(mathematics)), and [circle inversion](/source/circle_inversion) in [geometry](/source/geometry); [complementation](/source/complement_(set_theory)) in [set theory](/source/set_theory); and [reciprocal cipher](/source/reciprocal_cipher)s such as the [ROT13](/source/ROT13) transformation and the [Beaufort](/source/Beaufort_cipher) [polyalphabetic cipher](/source/polyalphabetic_cipher).

The [composition](/source/Function_composition) {{math|''g'' ∘ ''f''}} of two involutions {{math|''f''}} and {{math|''g''}} is an involution if and only if they [commute](/source/Commutative_property): {{math|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>

== Involutions on finite sets ==
The number of involutions, including the identity involution, on a set with {{math|1=''n'' = 0, 1, 2, ...}} elements is given by a [recurrence relation](/source/recurrence_relation) found by [Heinrich August Rothe](/source/Heinrich_August_Rothe) in 1800:
: <math>a_0 = a_1 = 1</math> and <math>a_n = a_{n - 1} + (n - 1)a_{n-2}</math> for <math>n > 1.</math>
The first few terms of this sequence are [1](/source/1_(number)), 1, [2](/source/2_(number)), [4](/source/4_(number)), [10](/source/10_(number)), [26](/source/26_(number)), [76](/source/76_(number)), [232](/source/232_(number)) {{OEIS|id=A000085}}; these numbers are called the [telephone numbers](/source/Telephone_number_(mathematics)), and they also count the number of [Young tableau](/source/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](/source/The_Art_of_Computer_Programming), Volume 3: Sorting and Searching
 | year = 1973
}}</ref>
The number {{math|''a''{{sub|''n''}}}} can also be expressed by non-recursive formulas, such as the sum
<math display="block">a_n = \sum_{m=0}^{\lfloor \frac{n}{2} \rfloor} \frac{n!}{2^m m! (n-2m)!} .</math>

The number of fixed points of an involution on a finite set and its [number of elements](/source/Cardinality) have the same [parity](/source/parity_(mathematics)). Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an [odd number](/source/odd_number) of elements has at least one [fixed point](/source/Fixed_point_(mathematics)). This can be used to prove [Fermat's two squares theorem](/source/Fermat's_theorem_on_sums_of_two_squares).<ref>
{{citation
 | last = Zagier | first = D. | author-link = Don Zagier
 | doi = 10.2307/2323918
 | issue = 2
 | journal = [American Mathematical Monthly](/source/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
 | jstor = 2323918
}}.</ref>

== Involution throughout the fields of mathematics ==

=== Real-valued functions ===

The [graph](/source/Graph_of_a_function) of an involution (on the real numbers) is [symmetric](/source/Reflection_symmetry) 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](/source/Euclidean_space) is [reflection](/source/Reflection_(mathematics)) through a [plane](/source/plane_(mathematics)). Performing a reflection twice brings a point back to its original coordinates.

Another involution is [reflection through the origin](/source/reflection_through_the_origin); not a reflection in the above sense, and so, a distinct example.

These transformations are examples of [affine involution](/source/affine_involution)s.

=== Projective geometry ===
An involution is a [projectivity](/source/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](/source/Cambridge_University_Press) via [Internet Archive](/source/Internet_Archive)</ref>{{rp|24}}
* Any projectivity that interchanges two points is an involution.
* The three pairs of opposite sides of a [complete quadrangle](/source/complete_quadrangle) meet any line (not through a vertex) in three pairs of an involution. More generally, given four points and any line not through one of them, there exists a projective involution which exchanges any point on this line with the second intersection of the (possibly degenerate) conic through the given point and the four others. This result has been called [Desargues](/source/Desargues)'s Involution Theorem.<ref>[J. V. Field](/source/Judith_V._Field) and [J. J. Gray](/source/Jeremy_Gray_(mathematician)) (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](/source/Pappus_of_Alexandria).<ref>Ivor Thomas (editor) (1980) ''Selections Illustrating the History of Greek Mathematics'', Volume II, number 362 in the [Loeb Classical Library](/source/Loeb_Classical_Library) (Cambridge and London: Harvard and Heinemann), pp. 610&ndash;3</ref>
* If an involution has one [fixed point](/source/fixed_point_(mathematics)) and is not the identity, it has another, and consists of the correspondence between [harmonic conjugates](/source/projective_harmonic_conjugate) 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](/source/correlation_(projective_geometry)) of period 2.<ref>[H. S. M. Coxeter](/source/H._S._M._Coxeter) (1969) ''Introduction to Geometry'', pp. 244–8, [John Wiley & Sons](/source/John_Wiley_%26_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](/source/matrix_(mathematics)) {{math|''T''}}. Every matrix has a [transpose](/source/transpose), obtained by swapping rows for columns. This transposition is an involution on the set of matrices.  Since elementwise [complex conjugation](/source/complex_conjugation) is an independent involution, the [conjugate transpose](/source/conjugate_transpose) or [Hermitian adjoint](/source/Hermitian_adjoint) is also an involution.

The definition of involution extends readily to [modules](/source/module_(mathematics)). Given a module {{math|''M''}} over a [ring](/source/ring_(mathematics)) {{math|''R''}}, an {{math|''R''}} [endomorphism](/source/endomorphism) {{math|''f''}} of {{math|''M''}} is called an involution if {{math|{{itco|''f''}}<sup>2</sup>}} is the identity homomorphism on {{math|''M''}}.

[Involutions are related to idempotent](/source/Idempotent_element_(ring_theory))s; if {{math|2}} is invertible then they [correspond](/source/bijection) in a one-to-one manner.

In [functional analysis](/source/functional_analysis), [Banach *-algebra](/source/Banach_*-algebra)s and [C*-algebra](/source/C*-algebra)s are special types of [Banach algebra](/source/Banach_algebra)s with involutions.

=== Quaternion algebra, groups, semigroups ===
In a [quaternion algebra](/source/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](/source/antidistributive). It also appears in [groups](/source/group_(mathematics)) 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](/source/semigroup_with_involution), of which there are natural examples that are not groups, for example square matrix multiplication (i.e. the [full linear monoid](/source/full_linear_monoid)) with [transpose](/source/transpose) as the involution.

=== Ring theory ===
{{details|*-algebra}}
In [ring theory](/source/ring_theory), the word ''involution'' is customarily taken to mean an [antihomomorphism](/source/antihomomorphism) that is its own inverse function.
Examples of involutions in common rings include the [complex conjugation](/source/complex_conjugation) on the [complex plane](/source/complex_plane), its equivalent in the [split-complex number](/source/split-complex_number)s, and the transpose operation in a matrix ring.

When ''R'' is a commutative ring and ''A'' is an [algebra over ''R''](/source/algebra_over_a_ring), then an anti-involution &sigma; on A is a ''standard involution'' if it leaves ''R'' fixed in ''A'' and satisfies
:<math>\forall a \in A \ \  \sigma(a) + a \in R \ \ \text{and}\ \  a \sigma(a) \in R.</math><ref>Alexander J. Hahn (1994) ''Quadratic algebras, Clifford algebras, and Arithmetic Witt Groups'', page 77, Universitext, Springer, {{isbn|0-387-94110-X}}</ref><ref>{{cite book | last=Knus | first=Max-Albert | authorlink=Max-Albert Knus | title=Quadratic and Hermitian forms over Rings | series=Grundlehren der Mathematischen Wissenschaften | volume=294 | location=Berlin etc. | publisher=[Springer-Verlag](/source/Springer-Verlag) | year=1991 | isbn=3-540-52117-8 | zbl=0756.11008 | page=3 }}</ref>

=== Group theory ===
In [group theory](/source/group_theory), an element of a [group](/source/group_(mathematics)) is an involution if it has [order](/source/order_(group_theory)) 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](/source/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](/source/permutation_group)''. By the end of the 19th century, ''group'' was defined more broadly, and accordingly so was ''involution''.

A [permutation](/source/permutation) is an involution if and only if it can be written as a finite product of disjoint [transposition](/source/transposition_(mathematics))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](/source/classification_of_finite_simple_groups).

An element {{math|''x''}} of a group {{math|''G''}} is called [strongly real](/source/strongly_real_element) 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](/source/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 [regular polyhedra](/source/Platonic_solid) and their [generalizations to higher dimensions](/source/regular_polytope).

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

Generally in [non-classical logic](/source/non-classical_logic)s, negation that satisfies the law of double negation is called ''involutive''. In [algebraic semantics](/source/Algebraic_semantics_(mathematical_logic)), such a negation is realized as an involution on the algebra of [truth value](/source/truth_value)s. Examples of logics that have involutive negation are Kleene and Bochvar [three-valued logic](/source/three-valued_logic)s, [Łukasiewicz many-valued logic](/source/%C5%81ukasiewicz_logic), the [fuzzy logic](/source/fuzzy_logic) '[involutive monoidal t-norm logic](/source/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](/source/t-norm_fuzzy_logics).

The involutiveness of negation is an important characterization property for logics and the corresponding [varieties of algebras](/source/variety_(universal_algebra)). For instance, involutive negation characterizes [Boolean algebra](/source/Boolean_algebra_(structure))s among [Heyting algebra](/source/Heyting_algebra)s. Correspondingly, classical [Boolean logic](/source/classical_logic) arises by adding the law of double negation to [intuitionistic logic](/source/intuitionistic_logic). The same relationship holds also between [MV-algebra](/source/MV-algebra)s and [BL-algebra](/source/BL_(logic))s (and so correspondingly between [Łukasiewicz logic](/source/%C5%81ukasiewicz_logic) and fuzzy logic [BL](/source/BL_(logic))), IMTL and [MTL](/source/Monoidal_t-norm_logic), and other pairs of important varieties of algebras (respectively, corresponding logics).

In the study of [binary relation](/source/binary_relation)s, every relation has a [converse relation](/source/converse_relation). Since the converse of the converse is the original relation, the conversion operation is an involution on the [category of relations](/source/category_of_relations). Binary relations are [ordered](/source/partial_order) through [inclusion](/source/inclusion_(set_theory)). While this ordering is reversed with the [complementation](/source/complementation_(mathematics)) involution, it is preserved under conversion.

=== Computer science ===
The [XOR](/source/XOR) [bitwise operation](/source/bitwise_operation) with a given value for one parameter is an involution on the other parameter. XOR [masks](/source/Mask_(computing)) 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](/source/bitwise_NOT) operation which is XOR with an all-ones value, and [stream cipher](/source/stream_cipher) [encryption](/source/encryption), which is an XOR with a secret [keystream](/source/keystream).

This predates binary computers; practically all mechanical cipher machines implement a [reciprocal cipher](/source/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](/source/Legendre_transformation), which converts between the [Lagrangian](/source/Lagrangian_mechanics) and [Hamiltonian](/source/Hamiltonian_mechanics), is an involutive operation. 

Integrability, a central notion of physics and in particular the subfield of [integrable system](/source/integrable_system)s, is closely related to involution, for example in context of [Kramers–Wannier duality](/source/Kramers%E2%80%93Wannier_duality).

== See also ==
* [Atbash](/source/Atbash)
* [Automorphism](/source/Automorphism)
* [Idempotence](/source/Idempotence)
* [ROT13](/source/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 |s2cid=45639619 }}
* {{citation | last1=Knus | first1=Max-Albert | author1-link=Max-Albert Knus |last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Rost | first3=Markus | author3-link=Markus Rost | last4=Tignol | first4=Jean-Pierre | author4-link=Jean-Pierre Tignol |  title=The Book of Involutions | others=With a preface by [J. Tits](/source/Jacques_Tits) | zbl=0955.16001 | series=Colloquium Publications | publisher=[American Mathematical Society](/source/American_Mathematical_Society) | volume=44 | location=Providence, RI | year=1998 | isbn=0-8218-0904-0 }}
* {{springer|title=Involution|id=p/i052510}}

== External links ==
* {{Commonscatinline|Involution}}

Category:Algebraic properties of elements
Category:Functions and mappings

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