Cross-Relation is Congruence Relation
Theorem
Let $\struct {S, \circ}$ be a commutative semigroup with cancellable elements.
Let $\struct {C, \circ {\restriction_C} } \subseteq \struct {S, \circ}$ be the subsemigroup of cancellable elements of $\struct {S, \circ}$, where $\circ {\restriction_C}$ denotes the restriction of $\circ$ to $C$.
Let $\struct {S_1, \circ {\restriction_1} } \subseteq \struct {S, \circ}$ be a subsemigroup of $S$.
Let $\struct {S_2, \circ {\restriction_2} } \subseteq \struct {C, \circ {\restriction_C} }$ be a subsemigroup of $C$.
Let $\left({S_1 \times S_2, \oplus}\right)$ 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 cross-relation on $S_1 \times S_2$, defined as:
- $\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$
The cross-relation $\boxtimes$ is a congruence relation on $\struct {S_1 \times S_2, \oplus}$.
Proof
From Cross-Relation is Equivalence Relation we have that $\boxtimes$ is an equivalence relation.
We now need to show that:
| \(\ds \tuple {x_1, y_1}\) | \(\boxtimes\) | \(\ds \tuple {x_2, y_2}\) | ||||||||||||
| \(\, \ds \land \, \) | \(\ds \tuple {u_1, v_1}\) | \(\boxtimes\) | \(\ds \tuple {u_2, v_2}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\tuple {x_1, y_1} \oplus \tuple {u_1, v_1} }\) | \(\boxtimes\) | \(\ds \paren {\tuple {x_2, y_2} \oplus \tuple {u_2, v_2} }\) |
So:
| \(\ds \tuple {x_1, y_1}\) | \(\boxtimes\) | \(\ds \tuple {x_2, y_2}\) | ||||||||||||
| \(\, \ds \land \, \) | \(\ds \tuple {u_1, v_1}\) | \(\boxtimes\) | \(\ds \tuple {u_2, v_2}\) | By assumption | ||||||||||
| \(\ds \paren {x_1 \circ u_1} \circ \paren {y_2 \circ v_2}\) | \(=\) | \(\ds \paren {x_1 \circ y_2} \circ \paren {u_1 \circ v_2}\) | Commutativity and associativity of $\circ$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {x_2 \circ y_1} \circ \paren {u_2 \circ v_1}\) | By assumption | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {x_2 \circ u_2} \circ \paren {y_1 \circ v_1}\) | Commutativity and associativity of $\circ$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \tuple {x_1 \circ u_1, y_1 \circ v_1}\) | \(\boxtimes\) | \(\ds \tuple {x_2 \circ u_2, y_2 \circ v_2}\) | Definition of $\boxtimes$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {\tuple {x_1, y_1} \oplus \tuple {u_1, v_1} }\) | \(\boxtimes\) | \(\ds \paren {\tuple {x_2, y_2} \oplus \tuple {u_2, v_2} }\) | Definition of $\oplus$ |
So $\boxtimes$ is a congruence relation on $\struct {S \times C, \oplus}$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $\S 20$: The Integers