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 $$\forall x \in S: \phi \left({x}\right) = x$$.

Hence the result.

Comment
Thus, when discussing inverse completions of a commutative semigroup with cancellable elements, we can talk about the inverse completion of such a semigroup.