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

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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:

$\forall x,y \in R: (0_R \mathrel{\mathcal R} x) \land (0_R \mathrel{\mathcal R} y) \implies 0_R \mathrel{\mathcal R} (x \circ y)$.