Definition:Relation Compatible with Operation

Definition
Let $\left({S, \circ}\right)$ be a closed algebraic structure.

Let $\mathcal R$ be a relation on $S$.

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


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


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

Also see

 * Definition:Congruence Relation


 * Preorder Compatible with Operation is Congruence