# 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.

$\blacksquare$

## 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): $\S 20$: Theorem $20.5$: Corollary