Definition:Self-Inverse Element/Definition 2

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

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

$x$ is a self-inverse element of $\struct {S, \circ}$ :
 * $x$ is invertible

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

Also see

 * Equivalence of Definitions of Self-Inverse