Definition:Inverse of Subset

Monoid
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 in the set $X$.

Group
When $\left({G, \circ}\right)$ is a group, then $C = G$ as from the Cancellation Laws, all group elements are cancelable.

Thus every subset has an inverse. Since each element has a unique inverse, $X^{-1}$ can be defined as:


 * $X^{-1} = \left\{{x \in G: x^{-1} \in X}\right\}$ or, equivalently,
 * $X^{-1} = \left\{{x^{-1}: x \in X}\right\}$