# Definition:Cross-Relation on Natural Numbers

## Definition

Consider the commutative semigroup $\left({\N, +}\right)$ composed of the natural numbers $\N$ and 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 $\boxtimes$ be the relation on $\N \times \N$ defined as:

$\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$

This relation $\boxtimes$ 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

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