An involution is a function f:X\to X that, when applied twice, brings one back to the starting point.
In mathematics, an (anti)involution, or an involutory function, is a function f that is its own inverse,


for all x in the domain of f.^{[1]} For x in ℝ, this is often called Babbage's functional equation (1820).^{[2]}
General properties
Any involution is a bijection.
The identity map is a trivial example of an involution. Common examples in mathematics of more detailed involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other examples include circle inversion, rotation by a halfturn, and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher.
The number of involutions, including the identity involution, on a set with n = 0, 1, 2, … elements is given by a recurrence relation found by Heinrich August Rothe in 1800:

a_{0} = a_{1} = 1;

a_{n} = a_{n − 1} + (n − 1)a_{n − 2}, for n > 1.
The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (sequence A000085 in OEIS); these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells.^{[3]}
Involution throughout the fields of mathematics
Euclidean geometry
A simple example of an involution of the threedimensional Euclidean space is reflection against a plane. Performing a reflection twice brings a point back to its original coordinates.
Another is the socalled reflection through the origin; this is an abuse of language as it is not a reflection, though it is an involution.
These transformations are examples of affine involutions.
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.

If an involution has one fixed point, it has another, and consists of the correspondence between 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 of period 2. ^{[4]}
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 1s and −1s on the diagonal. 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_{1} and e_{2} are basis elements. There exists a linear transformation f which sends e_{1} to e_{2}, and sends e_{2} to e_{1}, and which is the identity on all other basis vectors. It can be checked that f(f(x))=x for all x in V. That is, f is an involution of V.
This definition extends readily to modules. Given a module M over a ring R, an R endomorphism f of M is called an involution if f ^{2} is the identity homomorphism on M.
Involutions are related to idempotents; if 2 is invertible then they correspond in a onetoone manner.
Quaternion algebra
In a quaternion algebra, an (anti)involution is defined by the following axioms: if we consider a transformation \begin{align} x &\mapsto f(x) \end{align} then an involution is

f(f(x))=x . An involution is its own inverse

An involution is linear: f(x_1+x_2)=f(x_1)+f(x_2) and f(\lambda x)=\lambda f(x)

f(x_1 x_2)=f(x_1) f(x_2)
An antiinvolution does not obey the last axiom but instead
Ring theory
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:
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 a^{2} = e, where e is the identity element.^{[5]}
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 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.
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 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 values. Examples of logics which have involutive negation are 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, for example, 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).
Computer science
The XOR bitwise operation with a given value for one parameter is also an involution. XOR 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.
References

^ Russell, Bertrand (1903), Principles of mathematics (2nd ed.), W. W. Norton & Company, Inc, pp. page 426,

^ Ritt, J. F. (1916). "On Certain Real Solutions of Babbage's Functional Equation". The Annals of Mathematics 17 (3): 113.

^ .

^ H. S. M. Coxeter (1969) Introduction to Geometry, pp 244–8, John Wiley & Sons

^ John S. Rose. "A Course on Group Theory". p. 10, section 1.13.

Todd A. Ell; Stephen J. Sangwine (2007), "Quaternion involutions and antiinvolutions", Computers & Mathematics with Applications 53 (1): 137–143,
Further reading
See also
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.