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

Definition
Let $(R,*,\circ)$ be a closed ringoid with an additive identity $0_R$.

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

Then $\mathcal R$ is compatible with the ringoid $(R,*,\circ)$ iff:
 * $\forall x,y \in R: (0_R \mathrel{\mathcal R} x) \land (0_R \mathrel{\mathcal R} y) \implies 0_R \mathrel{\mathcal R} (x \circ y)$.