Construction of Inverse Completion/Identity of Quotient Structure

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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 $c \in C$ be arbitrary.

Then:

$\left[\!\left[{\left({c, c}\right)}\right]\!\right]_\boxtimes$

is the identity of $T'$.


Proof

\(\displaystyle \left({x \circ c}\right) \circ y\) \(=\) \(\displaystyle x \circ \left({y \circ c}\right)\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left[\!\left[ {\left({x, y}\right)} \right]\!\right]_\boxtimes \oplus' \left[\!\left[{ \left({c, c}\right)} \right]\!\right]_\boxtimes\) \(=\) \(\displaystyle \left[\!\left[{ \left({x \circ c, y \circ c}\right)} \right]\!\right]_\boxtimes\) Definition of $\oplus'$
\(\displaystyle \) \(=\) \(\displaystyle \left[\!\left[{ \left({x, y}\right)} \right]\!\right]_\boxtimes\) Cancellability of elements of $C$

Hence the result, by definition of identity element.

$\blacksquare$


Sources