Cross-Relation is Congruence Relation

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Theorem

Let $\left({S, \circ}\right)$ be a commutative semigroup with cancellable elements.

Let $\left({C, \circ {\restriction_C}}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$, where $\circ {\restriction_C}$ denotes the restriction of $\circ$ to $C$.


Let $\left({S_1, \circ {\restriction_1}}\right) \subseteq \left({S, \circ}\right)$ be a subsemigroup of $S$.

Let $\left({S_2, \circ {\restriction_2}}\right) \subseteq \left({C, \circ {\restriction_C}}\right)$ be a subsemigroup of $C$.


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 $\boxtimes$ be the cross-relation on $S_1 \times S_2$, defined as:

$\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 \circ y_2 = x_2 \circ y_1$


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

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