Extension Theorem for Isomorphisms

Theorem
Let the following conditions be fulfilled:


 * Let $\left({S, \circ}\right)$ be a commutative semigroup with cancellable elements
 * Let $\phi$ be an isomorphism from $\left({S, \circ}\right)$ into a semigroup $\left({T, *}\right)$
 * Let $\left({S\,', \circ'}\right)$ be an inverse completion of $\left({S, \circ}\right)$
 * Let $\left({T\,', \circ'}\right)$ be an inverse completion of $\left({T, \circ}\right)$.

Then there is a unique isomorphism $\phi': S\,' \to T\,'$ extending $\phi$.

Proof
Let $C$ be the subsemigroup of cancellable elements of $S$.

It is proved that this is a semigroup by Cancellable Elements of a Semigroup.

The set of cancellable elements of $T$ is $\phi \left({C}\right)$.

By the Extension Theorem for Homomorphisms, there is:


 * A unique homomorphism $\phi'$ from $S\,'$ into $T\,'$ extending $\phi$, and
 * A unique homomorphism $\psi$ from $T\,'$ into $S\,'$ extending $\phi^{-1}$.

Therefore, $\psi \circ \phi'$ is an endomorphism of $S\,'$ whose restriction to $S$ is the identity monomorphism from $S$ into $S\,'$.

But by the Extension Theorem for Homomorphisms, the identity automorphism of $S\,'$ is the only endomorphism of $S\,'$ extending the identity monomorphism from $S$ into $S\,'$, so:
 * $\psi \circ \phi' = I_{S\,'}$

Similarly:
 * $\phi' \circ \psi = I_{T\,'}$

So by Bijection iff Left and Right Inverse, $\phi'$ is a bijection and therefore an isomorphism.