Definition:Relation Compatible with Operation

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

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

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