Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements

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Theorem

Let $\left({S, \circ}\right)$ be a commutative semigroup which has 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 $\left({S \times C, \oplus}\right)$ be the external direct product of $\left({S, \circ}\right)$ and $\left({C, \circ {\restriction_C}}\right)$, where $\oplus$ is the operation on $S \times C$ induced by $\circ$ on $S$ and $\circ {\restriction_C}$ on $C$.


Let $\boxtimes$ be the cross-relation on $S \times C$, defined as:

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

This cross-relation is a congruence relation on $S \times C$.


Let the quotient structure defined by $\boxtimes$ be:

$\left({T', \oplus'}\right) := \left({\dfrac {S \times C} \boxtimes, \oplus_\boxtimes}\right)$

where $\oplus_\boxtimes$ is the operation induced on $\dfrac {S \times C} \boxtimes$ by $\oplus$.


Let the mapping $\psi: S \to T'$ be defined as:

$\forall x \in S: \psi \left({x}\right) = \left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes$

Let $S'$ be the image $\psi \left[{S}\right]$ of $S$.


The set $C'$ of cancellable elements of the semigroup $S'$ is $\psi \left[{C}\right]$.


Proof

We have Morphism Property Preserves Cancellability.

Thus:

$c \in C \implies \psi \left({c}\right) \in C'$

So by Image of Subset under Relation is Subset of Image: Corollary 2:

$\psi \left[{C}\right] \subseteq C'$


From above, $\psi$ is an isomorphism.


Hence, also from Morphism Property Preserves Cancellability:

$c' \in C' \implies \psi^{-1} \left({c'}\right) \in C$

So by Image of Subset under Relation is Subset of Image: Corollary 3:

$\psi^{-1} \left[{C'}\right] \subseteq C$

Hence by definition of set equality:

$\psi \left[{C}\right] = C'$

$\blacksquare$


Sources