Diagonal Relation is Universally Congruent

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Theorem

The diagonal relation $\Delta_S$ on a set $S$ is universally congruent on $S$.


Proof

We have that the diagonal relation is an equivalence relation.


Let $\struct {S, \circ}$ be any algebraic structure.

\(\ds \) \(\) \(\ds x_1 \mathrel {\Delta_S} x_2 \land y_1 \mathrel {\Delta_S} y_2\)
\(\ds \) \(\leadsto\) \(\ds x_1 = x_2 \land y_1 = y_2\) Definition of Diagonal Relation
\(\ds \) \(\leadsto\) \(\ds x_1 \circ y_1 = x_2 \circ y_2\) as a consequence of equality
\(\ds \) \(\leadsto\) \(\ds \paren {x_1 \circ y_1} \mathrel {\Delta_S} \paren {x_2 \circ y_2}\) Definition of Diagonal Relation

$\Delta_S$ can therefore be described as universally congruent.

$\blacksquare$


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