Definition:Relation Conversely 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 inverse compatible with $\circ$ iff:


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


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