Cross-Relation is Congruence Relation

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


From Cross-Relation is Equivalence Relation we have that $\boxtimes$ is an equivalence relation.

We now need to show that:

\(\ds \tuple {x_1, y_1}\) \(\boxtimes\) \(\ds \tuple {x_2, y_2}\)
\(\, \ds \land \, \) \(\ds \tuple {u_1, v_1}\) \(\boxtimes\) \(\ds \tuple {u_2, v_2}\)
\(\ds \leadsto \ \ \) \(\ds \paren {\tuple {x_1, y_1} \oplus \tuple {u_1, v_1} }\) \(\boxtimes\) \(\ds \paren {\tuple {x_2, y_2} \oplus \tuple {u_2, v_2} }\)


\(\ds \tuple {x_1, y_1}\) \(\boxtimes\) \(\ds \tuple {x_2, y_2}\)
\(\, \ds \land \, \) \(\ds \tuple {u_1, v_1}\) \(\boxtimes\) \(\ds \tuple {u_2, v_2}\) 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 $\struct {S \times C, \oplus}$.