Real-valued functions The
graph of an involution (on the real numbers) is
symmetric across the line . This is due to the fact that the inverse of any
general function will be its reflection over the line . This can be seen by "swapping" with . If, in particular, the function is an
involution, then its graph is its own reflection. Some basic examples of involutions include the functions \begin{alignat}{1} f(x) &= a-x \; , \\ f(x) &= \frac{b}{x-a}+a \end{alignat}Besides, we can construct an involution by wrapping an involution in a bijection and its inverse (h^{-1} \circ g \circ h). For instance :\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}
Euclidean geometry A simple example of an involution of the three-dimensional
Euclidean space is
reflection through a
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 involutions.
Projective geometry An involution is a
projectivity of period 2, that is, a projectivity that interchanges pairs of points. • 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. 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's Involution Theorem. 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. • If an involution has one
fixed point and is not the identity, 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". In the context of projectivities, fixed points are called
double points.
Linear algebra In linear algebra, an involution is a linear operator on a vector space, such that . Except for in characteristic 2, such operators are diagonalizable for a given basis with just s and 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 is chosen, and that and are basis elements. There exists a linear transformation that sends to , and sends to , and that is the identity on all other basis vectors. It can be checked that for all in . That is, is an involution of . For a specific basis, any linear operator can be represented by a
matrix . 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
modules. Given a module over a
ring , an
endomorphism of is called an involution if is the identity homomorphism on .
Involutions are related to idempotents; if is invertible then they
correspond in a one-to-one manner. In
functional analysis,
Banach *-algebras and
C*-algebras are special types of
Banach algebras with involutions.
Quaternion algebra, groups, semigroups In a
quaternion algebra, an (anti-)involution is defined by the following axioms: if we consider a transformation x \mapsto f(x) then it is an involution if • f(f(x))=x (it is its own inverse) • f(x_1+x_2)=f(x_1)+f(x_2) and f(\lambda x)=\lambda f(x) (it is linear) • f(x_1 x_2)=f(x_1) f(x_2) An anti-involution does not obey the last axiom but instead • f(x_1 x_2)=f(x_2) f(x_1) This former law is sometimes called
antidistributive. It also appears in
groups as . 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 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 include the
complex conjugation on the
complex plane, its equivalent in the
split-complex numbers, and the transpose operation in a matrix ring. When
R is a commutative ring and
A is an
algebra over R, then an anti-involution σ on A is a
standard involution if it leaves
R fixed in
A and satisfies :\forall a \in A \ \ \sigma(a) + a \in R \ \ \text{and}\ \ a \sigma(a) \in R.
Group theory In
group theory, an element of a
group is an involution if it has
order 2; that is, an involution is an element such that and , where is the
identity element. 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
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. An element of a group is called
strongly real if there is an involution with (where ).
Coxeter groups are groups generated by a set of involutions subject only to relations involving powers of pairs of elements of . 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: is equivalent to . 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 values. Examples of logics that have involutive negation are Kleene and Bochvar
three-valued logics,
Łukasiewicz many-valued logic, the
fuzzy 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
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
MV-algebras and
BL-algebras (and so correspondingly between
Łukasiewicz logic and fuzzy logic
BL), IMTL and
MTL, and other pairs of important varieties of algebras (respectively, corresponding logics). In the study of
binary relations, 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
ordered through
inclusion. While this ordering is reversed with the
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
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. Another involution used in computers is an order-2 bitwise permutation. For example, a color value stored as integers in the form , could exchange and , resulting in the form : .
Physics Legendre transformation, which converts between the
Lagrangian and
Hamiltonian, is an involutive operation. Integrability, a central notion of physics and in particular the subfield of
integrable systems, is closely related to involution, for example in context of
Kramers–Wannier duality. == See also ==