\begin{definition}[Definition:Self-Inverse Element/Definition 1]
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}} $x \circ x = e$.
\end{definition}