Equivalence Relation is Congruence for Right Operation

Theorem

Every equivalence is a congruence for the right operation.

Proof

Let $\mathcal R$ be an equivalence relation on the structure $\left({S, \rightarrow}\right)$.

Then:

$x_1 \rightarrow y_1 = y_1$
$x_2 \rightarrow y_2 = y_2$

Suppose $x_1 \mathop {\mathcal R} x_2 \land y_1 \mathop {\mathcal R} y_2$.

It follows directly that:

$\left({x_1 \rightarrow y_1}\right) \mathop {\mathcal R} \left({x_2 \rightarrow y_2}\right)$

$\blacksquare$