Definition:Self-Inverse Element

Definition
Let $\left({S, \circ}\right)$ be a monoid whose identity element is $e$.

Let $x\in S$ be an element.

Definition 1
$x$ is a self-inverse $x \circ x = e$.

Definition 2
$x$ is a self-inverse $x$ is invertible and $x = x^{-1}$, where $x^{-1}$ is the inverse of $x$.

Also known as
The definition is usually made in the context of a group.

Also see

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