Structure of Inverse Completion of Commutative Semigroup

Theorem
Let $\left({S, \circ}\right)$ be a commutative semigroup.

Let $\left ({C, \circ}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$.

Let $\left({T, \circ'}\right)$ be an inverse completion of $\left({S, \circ}\right)$.

Then:
 * $T = S \circ' C^{-1}$

where:
 * $C^{-1}$ is the inverse of $C$ in $T$
 * $S \circ' C^{-1}$ is the subset product of $S$ with $C^{-1}$.

Proof
Let $a \in C$.

Then:

Thus, as:
 * $C^{-1} \subseteq S \circ' C^{-1}$

and:
 * $S \subseteq S \circ' C^{-1}$

by Union is Smallest Superset it follows that:
 * $S \cup C^{-1} \subseteq S \circ' C^{-1}$

By the definition of the inverse completion:
 * $T = S \circ' C^{-1}$