Trivial Relation is Universally Congruent

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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:

\(\ds x_1, x_2, y_1, y_2\) \(\in\) \(\ds S\)
\(\ds \implies \ \ \) \(\ds x_1 \circ y_1, x_2 \circ y_2\) \(\in\) \(\ds S\) Definition of closed algebraic structure
\(\ds \implies \ \ \) \(\ds \left({x_1 \circ y_1}\right)\) \(\mathcal R\) \(\ds \left({x_2 \circ y_2}\right)\) Definition of Trivial Relation

$\blacksquare$


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