Inmathematics, aninvolution,involutory function, orself-inverse function[1] is afunctionf that is its owninverse,
for allx in thedomain off.[2] Equivalently, applyingf twice produces the original value.
Any involution is abijection.
Theidentity map is a trivial example of an involution. Examples of nontrivial involutions includenegation (x ↦ −x),reciprocation (x ↦ 1/x), andcomplex conjugation (z ↦z) inarithmetic;reflection, half-turnrotation, andcircle inversion ingeometry;complementation inset theory; andreciprocal ciphers such as theROT13 transformation and theBeaufortpolyalphabetic cipher.
Thecompositiong ∘f of two involutionsf andg is an involution if and only if theycommute:g ∘f =f ∘g.[3]
The number of involutions, including the identity involution, on a set withn = 0, 1, 2, ... elements is given by arecurrence relation found byHeinrich August Rothe in 1800:
The first few terms of this sequence are1, 1,2,4,10,26,76,232 (sequenceA000085 in theOEIS); these numbers are called thetelephone numbers, and they also count the number ofYoung tableaux with a given number of cells.[4]The numberan can also be expressed by non-recursive formulas, such as the sum
The number of fixed points of an involution on a finite set and itsnumber of elements have the sameparity. Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on anodd number of elements has at least onefixed point. This can be used to proveFermat's two squares theorem.[5]
Thegraph of an involution (on the real numbers) issymmetric across the liney =x. This is due to the fact that the inverse of anygeneral function will be its reflection over the liney =x. This can be seen by "swapping"x withy. If, in particular, the function is aninvolution, then its graph is its own reflection.
Some basic examples of involutions include the functionsBesides, we can construct an involution by wrapping an involutiong in a bijectionh and its inverse (). For instance :
A simple example of an involution of the three-dimensionalEuclidean space isreflection through aplane. Performing a reflection twice brings a point back to its original coordinates.
Another involution isreflection through the origin; not a reflection in the above sense, and so, a distinct example.
These transformations are examples ofaffine involutions.
An involution is aprojectivity of period 2, that is, a projectivity that interchanges pairs of points.[6]: 24
Another type of involution occurring in projective geometry is apolarity that is acorrelation of period 2.[9]
In linear algebra, an involution is a linear operatorT on a vector space, such thatT2 =I. Except for in characteristic 2, such operators are diagonalizable for a given basis with just1s and−1s on the diagonal of the corresponding matrix. If the operator is orthogonal (anorthogonal involution), it is orthonormally diagonalizable.
For example, suppose that a basis for a vector spaceV is chosen, and thate1 ande2 are basis elements. There exists a linear transformationf that sendse1 toe2, and sendse2 toe1, and that is the identity on all other basis vectors. It can be checked thatf(f(x)) =x for allx inV. That is,f is an involution ofV.
For a specific basis, any linear operator can be represented by amatrixT. Every matrix has atranspose, obtained by swapping rows for columns. This transposition is an involution on the set of matrices. Since elementwisecomplex conjugation is an independent involution, theconjugate transpose orHermitian adjoint is also an involution.
The definition of involution extends readily tomodules. Given a moduleM over aringR, anRendomorphismf ofM is called an involution iff2 is the identity homomorphism onM.
Involutions are related to idempotents; if2 is invertible then theycorrespond in a one-to-one manner.
Infunctional analysis,Banach *-algebras andC*-algebras are special types ofBanach algebras with involutions.
In aquaternion algebra, an (anti-)involution is defined by the following axioms: if we consider a transformation then it is an involution if
An anti-involution does not obey the last axiom but instead
This former law is sometimes calledantidistributive. It also appears ingroups as(xy)−1 = (y)−1(x)−1. Taken as an axiom, it leads to the notion ofsemigroup with involution, of which there are natural examples that are not groups, for example square matrix multiplication (i.e. thefull linear monoid) withtranspose as the involution.
Inring theory, the wordinvolution is customarily taken to mean anantihomomorphism that is its own inverse function.Examples of involutions in common rings:
Ingroup theory, an element of agroup is an involution if it hasorder 2; that is, an involution is an elementa such thata ≠e anda2 =e, wheree is theidentity element.[10]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 meanpermutation group. By the end of the 19th century,group was defined more broadly, and accordingly so wasinvolution.
Apermutation is an involution if and only if it can be written as a finite product of disjointtranspositions.
The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in theclassification of finite simple groups.
An elementx of a groupG is calledstrongly real if there is an involutiont with xt =x−1 (where xt =x−1 =t−1 ⋅x ⋅t).
Coxeter groups are groups generated by a setS of involutions subject only to relations involving powers of pairs of elements ofS. Coxeter groups can be used, among other things, to describe the possibleregular polyhedra and theirgeneralizations to higher dimensions.
The operation of complement inBoolean algebras is an involution. Accordingly,negation inclassical logic satisfies thelaw of double negation:¬¬A is equivalent toA.
Generally innon-classical logics, negation that satisfies the law of double negation is calledinvolutive. Inalgebraic semantics, such a negation is realized as an involution on the algebra oftruth values. Examples of logics that have involutive negation are Kleene and Bochvarthree-valued logics,Łukasiewicz many-valued logic, thefuzzy 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, int-norm fuzzy logics.
The involutiveness of negation is an important characterization property for logics and the correspondingvarieties of algebras. For instance, involutive negation characterizesBoolean algebras amongHeyting algebras. Correspondingly, classicalBoolean logic arises by adding the law of double negation tointuitionistic logic. The same relationship holds also betweenMV-algebras andBL-algebras (and so correspondingly betweenŁukasiewicz logic and fuzzy logicBL), IMTL andMTL, and other pairs of important varieties of algebras (respectively, corresponding logics).
In the study ofbinary relations, every relation has aconverse relation. Since the converse of the converse is the original relation, the conversion operation is an involution on thecategory of relations. Binary relations areordered throughinclusion. While this ordering is reversed with thecomplementation involution, it is preserved under conversion.
TheXORbitwise operation with a given value for one parameter is an involution on the other parameter. XORmasks 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 thebitwise NOT operation which is XOR with an all-ones value, andstream cipherencryption, which is an XOR with a secretkeystream.
This predates binary computers; practically all mechanical cipher machines implement areciprocal 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.[11]
Another involution used in computers is an order-2 bitwise permutation. For example, a color value stored as integers in the form(R,G,B), could exchangeR andB, resulting in the form(B,G,R):f(f(RGB)) = RGB,f(f(BGR)) = BGR.
Legendre transformation, which converts between theLagrangian andHamiltonian, is an involutive operation.
Integrability, a central notion of physics and in particular the subfield ofintegrable systems, is closely related to involution, for example in context ofKramers–Wannier duality.
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