Construction of Inverse Completion/Cartesian Product with Cancellable Elements

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 $C$.

That is:

$\forall \left({x, y}\right), \left({u, v}\right) \in S \times C: \left({x, y}\right) \oplus \left({u, v}\right) = \left({x \circ u, y \circ \restriction_C v}\right)$

Then $\left({S \times C, \oplus}\right)$ is a commutative semigroup.

Proof

By Cancellable Elements of a Semigroup, $\left({C, \circ \restriction_C}\right)$ is a subsemigroup of $\left({S, \circ}\right)$, where $\circ \restriction_C$ is the restriction of $\circ$ to $C$.

By Restriction of Operation Commutativity, as $\left({C, \circ \restriction_C}\right)$ is a substructure of a commutative structure, it is also commutative.

From:

• the external direct product preserves the nature of semigroups
• the external direct product preserves commutativity

we see that $\left({S \times C, \oplus}\right)$ is a commutative semigroup.

$\blacksquare$

Sources

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