\begin{definition}[Definition:Inverse of Subset/Monoid]
Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.
Let $C \subseteq S$ be the set of cancellable elements of $S$.
Let $X \subseteq C$.
Then the '''inverse''' of the subset $X$ is defined as:
:$X^{-1} = \set {y \in S: \exists x \in X: x \circ y = e_S}$
That is, it is the set of all the inverses of all the elements of $X$.
\end{definition}