\begin{definition}[Definition:Self-Inverse Element/Definition 2]
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 $\struct {S, \circ}$''' {{iff}}:
:$x$ is invertible
and:
:$x = x^{-1}$, where $x^{-1}$ is the inverse of $x$.
\end{definition}