Quotient Structure of Inverse Completion
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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.
This article, or a section of it, needs explaining. In particular: 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 $\struct {T, \circ'}$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
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