Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal 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 $C$.

Let $\mathcal R$ be the equivalence relation defined on $S \times C$ by:
 * $\left({x_1, y_1}\right) \ \mathcal R \ \left({x_2, y_2}\right) \iff x_1 \circ y_2 = x_2 \circ y_1$

Then:
 * $\forall c, d \in C: \left({c, c}\right) \ \mathcal R \ \left({d, d}\right)$

Proof
As $C \subseteq S$, from Cartesian Product of Subsets it follows that $C \times C \subseteq S \times C$.

Thus we need only consider elements $\left({x, y}\right)$ of $C \times C$.

Hence the result.