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 $C$.

Comment
In the context of the naturally ordered semigroup, the Unique Minus is defined:


 * $n \ominus m = p \iff m \circ p = n$

from which it can be seen that the above congruence can be understood as:


 * $\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 \circ y_2 = x_2 \circ y_1 \iff x_1 \ominus y_1 = x_2 \ominus y_2$

Thus this congruence defines an equivalence between pairs of elements which have the same Unique Minus (or, informally at this stage, difference).