Inverse Completion is Unique

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
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