Extension Theorem for Distributive Operations/Cancellability

Theorem
Then:
 * If $e$ is an identity for $\circ$, then $e$ is also an identity for $\circ'$.

Proof
By hypothesis, all the elements of $\struct {R, *}$ are cancellable.

Thus Inverse Completion of Commutative Semigroup is Abelian Group can be applied.

So $\struct {T, *}$ is an abelian group.

Let $e$ be the identity element of $\struct {R, \circ}$.

Then the restrictions to $R$ of the endomorphisms $\lambda_e: x \mapsto e \circ' x$ and $\rho_e: x \mapsto x \circ' e$ of $\struct {T, *}$ are monomorphisms.

But then $\lambda_e$ and $\rho_e$ are monomorphisms by the Extension Theorem for Homomorphisms, so $e$ is the identity element of $T$.