Trivial Relation is Universally Congruent

Theorem
The trivial relation $\mathcal R = S \times S$ on a set $S$ is universally congruent with every closed operation on $S$.

Proof
Let $\left({S, \circ}\right)$ be any algebraic structure which is closed for $\circ$.

By definition of trivial relation, $x \in S \land y \in S \implies x \mathcal R y$. So: