Definition:Inverse of Subset/Monoid

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Let $\left({S, \circ}\right)$ 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:

$X^{-1} = \left\{{y \in S: \exists x \in X: x \circ y = e_S}\right\}$

That is, it is the set of all the inverses of all the elements of $X$.