Construction of Inverse Completion/Cartesian Product with Cancellable Elements

Theorem
Let $$\left({S, \circ}\right)$$ be a commutative semigroup which has cancellable elements.

Let $$C \subseteq S$$ be the set of cancellable elements of $$S$$.

Let $$\left({S \times C, \oplus}\right)$$ be the external direct product of $$\left({S, \circ}\right)$$ and $$\left({C, \circ \restriction_C}\right)$$, where:
 * $$\circ \restriction_C$$ is the restriction of $\circ$ to $C \times C$, and
 * $$\oplus$$ is the operation on $S \times C$ induced by $\circ$ on $S$ and $\circ \restriction_C$ on $C$.

That is:

$$\forall \left({x, y}\right), \left({u, v}\right) \in S \times C: \left({x, y}\right) \oplus \left({u, v}\right) = \left({x \circ u, y \circ \restriction_C v}\right)$$

Then $$\left({S \times C, \oplus}\right)$$ is a commutative semigroup.

Proof
By Cancellable Elements of a Semigroup, $$\left({C, \circ \restriction_C}\right)$$ is a subsemigroup of $$\left({S, \circ}\right)$$, where $$\circ \restriction_C$$ is the restriction of $$\circ$$ to $$C$$.

By Restriction of Operation Commutativity, as $$\left({C, \circ \restriction_C}\right)$$ is a substructure of a commutative structure, it is also commutative.

From:


 * the external direct product preserves the nature of semigroups;
 * the external direct product preserves commutativity,

we see that $$\left({S \times C, \oplus}\right)$$ is a commutative semigroup.