\begin{definition}[Definition:Cross-Relation]
Let $\struct {S, \circ}$ be a commutative semigroup.
Let $\struct {S_1, \circ_{\restriction_1} }, \struct {S_2, \circ_{\restriction_2} }$ be subsemigroups of $S$, where $\circ_{\restriction_1}$ and $\circ_{\restriction_2}$ are the restrictions of $\circ$ to $S_1$ and $S_2$ respectively.
Let $\struct {S_1 \times S_2, \oplus}$ be the (external) direct product of $\struct {S_1, \circ_{\restriction_1} }$ and $\struct {S_2, \circ_{\restriction_2} }$, where $\oplus$ is the operation on $S_1 \times S_2$ induced by $\circ_{\restriction_1}$ on $S_1$ and $\circ_{\restriction_2}$ on $S_2$.
Let $\boxtimes$ be the relation on $S_1 \times S_2$ defined as:
:$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$
This relation $\boxtimes$ is referred to as the '''cross-relation on $\struct {S_1 \times S_2, \oplus}$'''.
\end{definition}