Inverse Completion is Commutative Semigroup

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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

From Subset Product defining Inverse Completion of Commutative Semigroup is Commutative Semigroup:

$S \circ' C^{-1}$ is a commutative semigroup.

From Structure of Inverse Completion of Commutative Semigroup:

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

$\blacksquare$


Sources