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 S x C: x1y2 = x2y1: 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 $$S$$.