Definition:Relation Compatible with Operation

Definition
Let $\mathcal R$ be a relation on an algebraic structure $\left({S, \circ}\right)$.

Then $\mathcal R$ is compatible with $\circ$ iff:


 * $\forall x, y, z \in S: x \mathop {\mathcal R} y \implies \left({x \circ z}\right) \mathop {\mathcal R} \left({y \circ z}\right)$


 * $\forall x, y, z \in S: x \mathop {\mathcal R} y \implies \left({z \circ x}\right) \mathop {\mathcal R} \left({z \circ y}\right)$

Also see

 * Congruence relation


 * Preorder Compatible with Operation is a Congruence