Inverse Completion is Commutative Semigroup
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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 $20.1: \ 2^\circ$