Equivalence Relation is Congruence for Left Operation
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Theorem
Every equivalence relation is a congruence for the left operation $\leftarrow$.
Proof
Let $\RR$ be an equivalence relation on the structure $\struct {S, \leftarrow}$.
Then:
- $x_1 \leftarrow y_1 = x_1$
- $x_2 \leftarrow y_2 = x_2$
Suppose $x_1 \mathrel \RR x_2 \land y_1 \mathrel \RR y_2$.
It follows directly that:
- $\paren {x_1 \leftarrow y_1} \mathrel \RR \paren {x_2 \leftarrow y_2}$
$\blacksquare$
Also se
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Example $11.4$