User:Dfeuer/Definition:Relation Compatible with Closed Ringoid with Zero

Definition

Let $\struct {R, *, \circ}$ be a closed ringoid with an additive identity $0_R$.

Let $\RR$ be a relation on $R$ which is compatible with $*$.

Then $\RR$ is compatible with the ringoid $\struct {R, *, \circ}$ if and only if:

$\forall x, y \in R: \paren {0_R \mathrel \RR x} \land \paren {0_R \mathrel \RR y] \implies 0_R \mathrel \RR \paren {x \circ y} }$.