Definition:Relation Strongly Compatible with Operation

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

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

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

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

That is, if and only if $\RR$ is compatible with $\circ$ and conversely compatible with $\circ$.

Also see

Linguistic Note

The term relation strongly compatible with operation was invented by $\mathsf{Pr} \infty \mathsf{fWiki}$.

As such, it is not generally expected to be seen in this context outside $\mathsf{Pr} \infty \mathsf{fWiki}$.