an (anti-)involution, or
an involutary function, is
a function f that
is its own inverse:
= x for all x in the domain of f.
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,
the ROT13 transformation,
and the Beaufort polyalphabetic
The number of involutions on a set with n =
0, 1, 2, ... elements is given by the recurrence
a0 = a1 =
an = an −
1 + (n − 1)an − 2,
for n > 1.
The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (sequence A000085 in OEIS).
throughout the fields of mathematics
A simple example of an involution of the
space is reflection against a plane.
Performing a reflection twice brings us back where we started.
Another is the so-called reflection through the origin; this is
of language, as it is an involution, but not a reflection.
These transformations are examples of affine involutions.
In linear algebra, an involution is a linear
operator T such that T2 = I.
Except for in characteristic 2, such operators are diagonalizable with 1's and
-1's on the diagonal. If the operator is orthogonal (an orthogonal involution), it is
The transpose of a square matrix is an involution
because the transpose of the transpose of a matrix is itself.
are related to idempotents; if 2 is invertible, (in a field
of characteristic other than 2), then they are equivalent.
In a Quaternion algebra, an (anti-)involution is
defined by following axioms: if we consider a transformation then a
§ f(f(x)) = x. An
involution is its own inverse
§ An involution is linear: f(x1 + x2)
= f(x1) + f(x2) and f(λx)
An anti-involution does not obey the last axiom but
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:
conjugation on the complex plane
§ multiplication by j in the split-complex
§ taking the transpose in
a matrix ring
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 a2 = e,
where e is the identity element.
Originally, this definition differed not at all from 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. The group
of bijections generated by an involution through composition, is isomorphic
A permutation is
an involution precisely if it can be written as a product of one or more
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.
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
to higher dimensions.
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 which
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, e.g., Kleene and Bochvar three-valued
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 e.g. in t-norm
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 betweenMV-algebras and BL-algebras (and so correspondingly
logic and fuzzy logic BL), IMTL and MTL,
and other pairs of important varieties of algebras (resp. corresponding