Definition:Self-Inverse Element

Definition

Let $\struct {S, \circ}$ be a monoid whose identity element is $e_S$.

Let $x \in S$ be an element of $S$.

Definition 1

$x$ is a self-inverse element of $\struct {S, \circ}$ if and only if $x \circ x = e_S$.

Definition 2

$x$ is a self-inverse element of $\struct {S, \circ}$ if and only if:

$x$ is invertible

and:

$x = x^{-1}$, where $x^{-1}$ is the inverse of $x$.

Also known as

The definition of a self-inverse element is usually made in the context of a group.

Some sources refer to such an element as an involution.

Also see

• Equivalence of Definitions of Self-Inverse
• Inverse in Monoid is Unique

• Results about self-inverse elements can be found here.