Restriction of Congruence Relation is Congruence

Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\RR$ be a congruence relation for $\circ$ on $S$.

Let $T \subseteq S$ be a subset of $S$.

Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.

Then $\RR {\restriction_T}$ is a congruence relation for $\circ_T$ on $T$.

Proof

\(\ds \forall x_1, x_2, y_1, y_2 \in T: \, \)

\(\ds \)

\(\)

\(\ds \paren {x_1 \mathrel {\RR {\restriction_T} } x_2} \land \paren {y_1 \mathrel {\RR {\restriction_T} } y_2}\)

\(\ds \leadsto \ \ \)

\(\ds \)

\(\)

\(\ds \paren {x_1 \mathrel \RR x_2} \land \paren {y_1 \mathrel \RR y_2}\)

Definition of Restriction of Relation

\(\ds \leadsto \ \ \)

\(\ds \)

\(\)

\(\ds \paren {x_1 \circ y_1} \mathrel \RR \paren {x_2 \circ y_2}\)

Definition of Congruence Relation: $\RR$ is a congruence relation for $\circ$ on $S$

\(\ds \leadsto \ \ \)

\(\ds \)

\(\)

\(\ds \paren {x_1 \circ_T y_1} \mathrel {\RR {\restriction_T} } \paren {x_2 \circ_T y_2}\)

Definition of Restriction of Relation, Definition of Restriction of Operation

Hence the result by definition of congruence relation.

$\blacksquare$