Inverse Completion is 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 $T = S \circ' C^{-1}$, and is a commutative semigroup.

Proof

$S \circ' C^{-1}$ is a commutative semigroup.
$T = S \circ' C^{-1}$

$\blacksquare$