Elements of Cross-Relation Equivalence Class

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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:

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


Let $\left[\!\left[{\left({x, y}\right)}\right]\!\right]_\boxtimes$ be the $\boxtimes$-equivalence class of $\left({x, y}\right)$, where $\left({x, y}\right) \in S_1 \times S_2$.


Then:

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

$(1): \quad \left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes = \left[\!\left[{\left({y \circ b, b}\right)}\right]\!\right]_\boxtimes \iff x = y$
$(2): \quad \left[\!\left[{\left({x \circ a, y \circ a}\right)}\right]\!\right]_\boxtimes = \left[\!\left[{\left({x, y}\right)}\right]\!\right]_\boxtimes$


Proof

\((1):\quad\) \(\displaystyle \left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes\) \(=\) \(\displaystyle \left[\!\left[{\left({y \circ b, b}\right)}\right]\!\right]_\boxtimes\)
\(\displaystyle \left({x \circ a, a}\right)\) \(\boxtimes\) \(\displaystyle \left({y \circ b, b}\right)\)
\(\displaystyle \iff \ \ \) \(\displaystyle x \circ a \circ b\) \(=\) \(\displaystyle y \circ b \circ a\) Definition of $\boxtimes$
\(\displaystyle \) \(=\) \(\displaystyle y \circ a \circ b\) Commutativity of $\circ$
\(\displaystyle \iff \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle y\) Cancellability of $a \circ b$


\((2):\quad\) \(\displaystyle \left({x \circ a}\right) \circ y\) \(=\) \(\displaystyle x \circ \left({a \circ y}\right)\) as $\circ$ is associative
\(\displaystyle \iff \ \ \) \(\displaystyle \left({x \circ a, y \circ a}\right)\) \(\boxtimes\) \(\displaystyle \left({x, y}\right)\) Definition of $\boxtimes$
\(\displaystyle \iff \ \ \) \(\displaystyle \left[\!\left[{\left({x \circ a, y \circ a}\right)}\right]\!\right]_\boxtimes\) \(=\) \(\displaystyle \left[\!\left[{\left({x, y}\right)}\right]\!\right]_\boxtimes\)

$\blacksquare$


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