Elements of Cross-Relation Equivalence Class
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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $\S 20$: The Integers