Definition:Relation Conversely Compatible with Operation

Definition

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

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

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

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

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

Also see

• Definition:Relation Compatible with Operation
• Definition:Relation Strongly Compatible with Operation

Linguistic Note

The term relation conversely 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.