Involution (mathematics)
In mathematics, an involution, involutory function, or self-inverse function[1] is a function f that is its own inverse,
- f(f(x)) = x
for all x in the domain of f.[2] Equivalently, applying f twice produces the original value.
General properties
Any involution is a bijection.
The
.The composition g ∘ f of two involutions f and g is an involution if and only if they commute: g ∘ f = f ∘ g.[3]
Involutions on finite sets
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:
- and for
The first few terms of this sequence are
The number of fixed points of an involution on a finite set and its
Involution throughout the fields of mathematics
Real-valued functions
The graph of an involution (on the real numbers) is symmetric across the line y = x. This is due to the fact that the inverse of any general function will be its reflection over the line y = x. This can be seen by "swapping" x with y. If, in particular, the function is an involution, then its graph is its own reflection. Some basic examples of involutions include the functions
Other nonlinear examples include (note the possibility of interpreting these as compositions):
Other elementary involutions are useful in solving functional equations.
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
These transformations are examples of affine involutions.
Projective geometry
An involution is a
- Any projectivity that interchanges two points is an involution.
- The three pairs of opposite sides of a Desargues's Involution Theorem.[7] 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.[8]
- 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". In the context of projectivities, fixed points are called double points.[6]: 53
Another type of involution occurring in projective geometry is a polarity that is a correlation of period 2.[9]
Linear algebra
In linear algebra, an involution is a linear operator T on a vector space, such that T2 = I. Except for in characteristic 2, such operators are diagonalizable for a given basis with just 1s and −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 e1 and e2 are basis elements. There exists a linear transformation f that sends e1 to e2, and sends e2 to e1, and that 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.
For a specific basis, any linear operator can be represented by a
The definition of involution extends readily to modules. Given a module M over a ring R, an R endomorphism f of M is called an involution if f2 is the identity homomorphism on M.
In
Quaternion algebra, groups, semigroups
In a quaternion algebra, an (anti-)involution is defined by the following axioms: if we consider a transformation then it is an involution if
- (it is its own inverse)
- and (it is linear)
An anti-involution does not obey the last axiom but instead
This former law is sometimes called
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:
- complex conjugation on the complex plane, and its equivalent in the split-complex numbers
- taking the transpose in a matrix ring.
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 a such that a ≠ e and a2 = e, where e is the identity 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 mean permutation group. By the end of the 19th century, group was defined more broadly, and accordingly so was involution.
A
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
Coxeter groups are groups generated by a set S of involutions subject only to relations involving powers of pairs of elements of S. 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 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, 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 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 (resp. corresponding logics).
In the study of
Computer science
The
Another example is a bit mask-and-shift function operating on colour values stored as integers, say in the form (R, G, B), that swaps R and B, resulting in the form (B, G, R): f(f(RGB)) = RGB, f(f(BGR)) = BGR.
The RC4 cryptographic cipher is an involution, as encryption and decryption operations use the same function.
Practically all mechanical cipher machines implement a
See also
References
- ISBN 0321307143, p. 165
- ISBN 9781440054167
- ISBN 9780817649982.
- ^
MR 0445948
- ^
MR 1041893.
- ^ a b A.G. Pickford (1909) Elementary Projective Geometry, Cambridge University Press via Internet Archive
- ^ J. V. Field and J. J. Gray (1987) The Geometrical Work of Girard Desargues, (New York: Springer), p. 54
- ^ Ivor Thomas (editor) (1980) Selections Illustrating the History of Greek Mathematics, Volume II, number 362 in the Loeb Classical Library (Cambridge and London: Harvard and Heinemann), pp. 610–3
- John Wiley & Sons
- ^ John S. Rose. "A Course on Group Theory". p. 10, section 1.13.
- ^ Greg Goebel. "The Mechanization of Ciphers" 2018.
Further reading
- Ell, Todd A.; Sangwine, Stephen J. (2007). "Quaternion involutions and anti-involutions". Computers & Mathematics with Applications. 53 (1): 137–143. S2CID 45639619.
- Knus, Max-Albert; Zbl 0955.16001
- "Involution", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
External links
- Media related to Involution at Wikimedia Commons