# Inverse Completion Theorem

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## Theorem

Every commutative semigroup containing cancellable elements admits an inverse completion.

## Proof

Let $\struct {S, \circ}$ be a commutative semigroup which has cancellable elements.

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

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

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

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $\S 20$: The Integers: Theorem $20.3$