Definition:Relation Compatible with Operation

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

Let $\RR$ be a relation on $S$.

Then $\RR$ is compatible with $\circ$ if and only if:

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

Also see

  • Results about compatible relations can be found here.