Construction of Inverse Completion/Properties of Quotient Structure

Theorem

Let $\left({S, \circ}\right)$ be a commutative semigroup which has cancellable elements.

Let $C \subseteq S$ be the set of cancellable elements of $S$.

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:

• $\circ \restriction_C$ is the restriction of $\circ$ to $C \times C$, and
• $\oplus$ is the operation on $S \times C$ induced by $\circ$ on $S$ and $\circ \restriction_C$ on $C$.

Let $\mathcal R$ be the congruence relation $\mathcal R$ defined on $\left({S \times C, \oplus}\right)$ by:

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

Let the quotient structure defined by $\mathcal R$ be:

$\displaystyle \left({T\,', \oplus'}\right) := \left({\frac {S \times C} {\mathcal R}, \oplus_\mathcal R}\right)$

where $\oplus_\mathcal R$ is the operation induced on $\displaystyle \frac {S \times C} {\mathcal R}$ by $\oplus$.

Identity of Quotient Structure

Let $c \in C$ be arbitrary.

Then:

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

is the identity of $T\,'$.

Invertible Elements in Quotient Structure

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

Generator for Quotient Structure

$T\,' = S\,' \cup \left({C\,'}\right)^{-1}$ is a generator for the semigroup $T\,'$.

Quotient Structure is Inverse Completion

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

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 20$