Inverse Completion is Unique
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Theorem
An inverse completion of a commutative semigroup is unique up to isomorphism.
Proof
Let $T$ and $T'$ both be inverse completions of a commutative semigroup $S$ having cancellable elements.
Then from the Extension Theorem for Isomorphisms, there is a unique isomorphism $\phi: T \to T'$ satisfying $\forall x \in S: \map \phi x = x$.
Hence the result.
$\blacksquare$
Comment
Thus, when discussing inverse completions of a commutative semigroup with cancellable elements, we can talk about the inverse completion of such a semigroup.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $\S 20$: The Integers: Theorem $20.5$: Corollary