Construction of Inverse Completion/Properties of Quotient Structure

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Let $\struct {S, \circ}$ be a commutative semigroup which has 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 $\struct {S \times C, \oplus}$ be the external direct product of $\struct {S, \circ}$ and $\struct {C, \circ {\restriction_C} }$, 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:

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

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

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

$\struct {T', \oplus'} := \struct {\dfrac {S \times C} \boxtimes, \oplus_\boxtimes}$

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

Identity of Quotient Structure

Let $c \in C$ be arbitrary.


$\eqclass {\tuple {c, c} } \boxtimes$

is the identity of $T'$.

Invertible Elements in Quotient Structure

Every cancellable element of $S'$ is invertible in $T'$.

Generator for Quotient Structure

$T' = S' \cup \paren {C'}^{-1}$ is a generator for the semigroup $T'$.

Quotient Structure is Inverse Completion

$T'$ is an inverse completion of its subsemigroup $S'$.