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, $$x \in S \land y \in S \Longrightarrow x \mathcal{R} y$$. So: