Cross-Relation is Equivalence Relation

Theorem
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be a commutative semigroups.

Let $\left({S \times T, \oplus}\right)$ be the external direct product of $\left({S, \circ}\right)$ and $\left({T, *}\right)$.

Let $\mathcal R$ be the cross-relation on $S \times T$, defined as:
 * $\left({x_1, y_1}\right) \mathop {\mathcal R} \left({x_2, y_2}\right) \iff x_1 \circ y_2 = x_2 \circ y_1$

Then $\mathcal R$ is an equivalence relation on $\left({S \times T, \oplus}\right)$.

Reflexivity

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

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.