Definition:Cross-Relation

Definition

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 $\struct {S_1 \times S_2, \oplus}$.

Natural Numbers

When developing the definition of integers, a cross-relation is often specified directly on the 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 $\left({\N \times \N, \oplus}\right)$.

Also see

• Cross-Relation is Equivalence Relation
• Results about cross-relations can be found here.

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 by which to refer to it.