Definition:Cross-Relation on Natural Numbers

Definition
Let $\left({\N, +}\right)$ be the semigroup or natural numbers under addition.

Let $\left({\N \times \N, \oplus}\right)$ be the (external) direct product of $\left({\N, +}\right)$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$.

Let $\mathcal R$ be the relation on $\N \times \N$ defined as:


 * $\left({x_1, y_1}\right) \mathop {\mathcal R} \left({x_2, y_2}\right) \iff x_1 \circ y_2 = x_2 \circ y_1$

This relation $\mathcal R$ is referred to as the cross-relation on $\left({\N \times \N, \oplus}\right)$.

Note on Terminology
The name for the definition of this relation on such an external direct product has been coined specifically for ProofWiki.

This relation occurs sufficiently frequently in the context of inverse completions that it needs a compact name to refer to it.