Definition:Relation Strongly Compatible with Operation

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

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

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


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


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

That is, iff $\mathcal R$ is compatible with $\circ$ and inverse compatible with $\circ$.