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.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965): $\S 20$: Theorem $20.5$