# Inverse Completion Theorem

## Theorem

Every commutative semigroup containing cancellable elements admits an inverse completion.

## Proof

Let $\left({S, \circ}\right)$ be a commutative semigroup which has cancellable elements.

From Construction of Inverse Completion, we can construct an inverse completion $\left({T', \oplus'}\right)$ of $\left({S', \oplus'}\right)$, which is an isomorphic copy of $S$ under the mapping $\psi: S \to S'$.

By the Embedding Theorem, there exists a semigroup $\left({T, \circ}\right)$ which contains $\left({S, \circ}\right)$, and an isomorphism $\Psi$ from $\left({T, \circ}\right)$ to $\left({T', \oplus'}\right)$ which extends $\psi$.

Thus $T$ is an inverse completion of $S$.

$\blacksquare$