Elements of Cross-Relation Equivalence Class

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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:

$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$


Let $\eqclass {\tuple {x, y} } \boxtimes$ be the $\boxtimes$-equivalence class of $\tuple {x, y}$, where $\tuple {x, y} \in S_1 \times S_2$.


Then:

$\forall x, y \in S_1, a, b \in S_2:$

$(1): \quad \eqclass {\tuple {x \circ a, a} } \boxtimes = \eqclass {\tuple {y \circ b, b} } \boxtimes \iff x = y$
$(2): \quad \eqclass {\tuple {x \circ a, y \circ a} } \boxtimes = \eqclass {\tuple {x, y} } \boxtimes$


Proof

\(\text {(1)}: \quad\) \(\ds \eqclass {\tuple {x \circ a, a} } \boxtimes\) \(=\) \(\ds \eqclass {\tuple {y \circ b, b} } \boxtimes\)
\(\ds \tuple {x \circ a, a}\) \(\boxtimes\) \(\ds \tuple {y \circ b, b}\)
\(\ds \leadstoandfrom \ \ \) \(\ds x \circ a \circ b\) \(=\) \(\ds y \circ b \circ a\) Definition of $\boxtimes$
\(\ds \) \(=\) \(\ds y \circ a \circ b\) Commutativity of $\circ$
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(=\) \(\ds y\) Cancellability of $a \circ b$


\(\text {(2)}: \quad\) \(\ds \paren {x \circ a} \circ y\) \(=\) \(\ds x \circ \paren {a \circ y}\) as $\circ$ is associative
\(\ds \leadstoandfrom \ \ \) \(\ds \tuple {x \circ a, y \circ a}\) \(\boxtimes\) \(\ds \tuple {x, y}\) Definition of $\boxtimes$
\(\ds \leadstoandfrom \ \ \) \(\ds \eqclass {\tuple {x \circ a, y \circ a} } \boxtimes\) \(=\) \(\ds \eqclass {\tuple {x, y} } \boxtimes\)

$\blacksquare$


Sources

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