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, $$x \in S \land y \in S \implies x \mathcal R y$$. So:

$$ $$ $$