Definition:Relation Compatible with Operation

Definition
Let $\struct {S, \circ}$ be a closed algebraic structure.

Let $\RR$ be a relation on $S$.

Then $\RR$ is compatible with $\circ$ :


 * $\forall x, y, z \in S: x \mathrel \RR y \implies \paren {x \circ z} \mathrel \RR \paren {y \circ z}$


 * $\forall x, y, z \in S: x \mathrel \RR y \implies \paren {z \circ x} \mathrel \RR \paren {z \circ y}$

Also see

 * Definition:Congruence Relation


 * Equivalence Relation is Congruence iff Compatible with Operation
 * Preordering of Products under Operation Compatible with Preordering