Construction of Inverse Completion/Equivalence Relation

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 $C$.

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

is an equivalence relation on $\left({S \times C, \oplus}\right)$.

Reflexivity

 * $x_1 \circ y_1 = x_1 \circ y_1 \implies \left({x_1, y_1}\right) \mathcal R \left({x_1, y_1}\right)$

So $\mathcal R$ is a reflexive relation.

Symmetry
So $\mathcal R$ is a symmetric relation.

Transitivity
So $\mathcal R$ is a transitive relation.

All the criteria are therefore seen to hold for $\mathcal R$ to be an equivalence relation.