Definition:Involution (Mapping)

Definition

Let $A$ be a set.

Let $f: A \to A$ be a mapping on $A$.

Definition 1

$f$ is an involution if and only if:

$\forall x \in A: \map f {\map f x} = x$

That is:

$f \circ f = I_A$

where $I_A$ denotes the identity mapping on $A$.

Definition 2

$f$ is an involution if and only if:

$\forall x, y \in A: \map f x = y \implies \map f y = x$

Definition 3

$f$ is an involution if and only if $f$ is both a bijection and a symmetric relation.

That is, if and only if $f$ is a bijection such that:

$f = f^{-1}$

Also known as

An involution is also known as an involutive mapping or an involutive function.

An involutive mapping can also be found described as self-inverse.

Also see

• Equivalence of Definitions of Involutive Mapping
• Mapping is Involution iff Bijective and Symmetric

• Results about involutions can be found here.

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): involution