Inverse Completion Theorem

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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 $S$.

$\blacksquare$


Sources