\begin{definition}[Definition:Cross-Relation on Natural Numbers]
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 $\left({\N \times \N, \oplus}\right)$'''.
\end{definition}