Equivalence Relation is Congruence for Right Operation
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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
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Example $11.4$