Definition:Cross-Relation

Definition
Let $\left({S, \circ}\right)$ be a commutative semigroups.

Let $\left({S_1, \circ_{\restriction_1}}\right), \left({S_2, \circ_{\restriction_2}}\right)$ 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 $\left({S_1 \times S_2, \oplus}\right)$ be the (external) direct product of $\left({S_1, \circ_{\restriction_1}}\right)$ and $\left({S_2, \circ_{\restriction_2}}\right)$, 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 $\mathcal R$ be the relation on $S_1 \times S_2$ 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 $\left({S_1 \times S_2, \oplus}\right)$.

Natural Numbers
When developing the definition of integers, a cross-relation is often specified directly on the natural numbers:

Also see

 * Cross-Relation is Equivalence Relation

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.