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:


 * $\forall x_1, x_2, y_1, y_2 \in S: x_1 \mathcal R x_2 \land y_1 \mathcal R y_2 \implies \left({x_1 \circ y_1}\right) \mathcal R \left({x_2 \circ y_2}\right)$

This is alternatively given as:


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


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

These definitions are equivalent.

The former is usually more often seen when the relation in question is an equivalence, while the latter tends to be used in the context of orderings.

Also see

 * Congruence relation