Equivalence Relation is Congruence for Right Operation

Theorem

Every equivalence relation is a congruence for the right operation $\rightarrow$.

Proof

Let $\RR$ be an equivalence relation on the structure $\struct {S, \rightarrow}$.

Then:

$x_1 \rightarrow y_1 = y_1$

$x_2 \rightarrow y_2 = y_2$

Suppose $x_1 \mathrel \RR x_2 \land y_1 \mathrel \RR y_2$.

It follows directly that:

$\paren {x_1 \rightarrow y_1} \mathrel \RR \paren {x_2 \rightarrow y_2}$

$\blacksquare$

Also see

• Equivalence Relation is Congruence for Left Operation

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Example $11.4$