Quotient Structure of Inverse Completion

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

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


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

Then the mapping $g: \paren {S \times C} / \RR_f \to T$ defined by $\map g {\eqclass {x, y} {\RR_f} } = x \circ' y^{-1}$,

where $\paren {S \times C} / \RR_f$ is a quotient structure, is an isomorphism.

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

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

By Quotient Theorem for Epimorphisms, the proof follows.