Quotient Structure of Inverse Completion

Theorem
Let $$\left({T, \circ'}\right)$$ be an inverse completion of a commutative semigroup $$\left({S, \circ}\right)$$, where $$C$$ is the set of cancellable elements of $$S$$.

Let $$f: S \times C: T$$ be the mapping defined as:


 * $$\forall x \in S, y \in C: f \left({x, y}\right) = x \circ' y^{-1}$$

Then the mapping $$g: \left({S \times C}\right) / \mathcal R_f \to T$$ defined by $$g \left({\left[\!\left[{x, y}\right]\!\right]_{\mathcal R_f}}\right) = x \circ' y^{-1}$$,

where $$\left({S \times C}\right) / \mathcal R_f$$ is a quotient structure, is an isomorphism.

Proof
$$T$$ is commutative, from Inverse Completion Commutative Semigroup.

The mapping $$f \left({x, y}\right) = x \circ' y^{-1}$$ is an epimorphism from the cartesian product of $$\left({S, \circ}\right)$$ and $$\left({C, \circ \restriction_C}\right)$$ onto $$\left({T, \circ'}\right)$$.

By the Quotient Theorem for Epimorphisms, the proof follows.