Construction of Inverse Completion

Theorem
This page consists of a series of linked theorems, each of which builds towards one result.

To access the proofs for the individual theorems, click on the links which form the titles of each major section.

Initial Definitions
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 $C$.