Trivial Relation is Universally Congruent

Theorem

The trivial relation $\RR = S \times S$ on a set $S$ is universally congruent on $S$.

Proof

Let $\struct {S, \circ}$ be any algebraic structure which is closed for $\circ$.

By definition of trivial relation:

$x \in S \land y \in S \implies x \mathrel \RR y$

So:

\(\ds x_1, x_2, y_1, y_2\)

\(\in\)

\(\ds S\)

\(\ds \leadsto \ \ \)

\(\ds x_1 \circ y_1, x_2 \circ y_2\)

\(\in\)

\(\ds S\)

Definition of Closed Algebraic Structure

\(\ds \leadsto \ \ \)

\(\ds \paren {x_1 \circ y_1}\)

\(\RR\)

\(\ds \paren {x_2 \circ y_2}\)

Definition of Trivial Relation

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Example $11.3$