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:

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

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

Also see

• Congruence relation

• Preorder Compatible with Operation is a Congruence

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 15$