Inverse Completion is Commutative Semigroup

Theorem

Let $\struct {S, \circ}$ be a commutative semigroup.

Let $\struct {C, \circ} \subseteq \struct {S, \circ}$ be the subsemigroup of cancellable elements of $\struct {S, \circ}$.

Let $\struct {T, \circ'}$ be an inverse completion of $\struct {S, \circ}$.


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

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $\S 20$: The Integers: Theorem $20.1: \ 2^\circ$
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