Definition:Relation Compatible with Operation

Definition
Let $\struct {S, \circ}$ be a closed algebraic structure.

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

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


 * $\forall x, y, z \in S: x \mathrel {\mathcal R} y \implies \paren {x \circ z} \mathrel {\mathcal R} \paren {y \circ z}$


 * $\forall x, y, z \in S: x \mathrel {\mathcal R} y \implies \paren {z \circ x} \mathrel {\mathcal R} \paren {z \circ y}$

Also see

 * Definition:Congruence Relation


 * Preordering of Products under Operation Compatible with Preordering