Inverse Completion Theorem

Theorem

Every commutative semigroup containing cancellable elements admits an inverse completion.


Proof

Let $\struct {S, \circ}$ be a commutative semigroup which has cancellable elements.


From Construction of Inverse Completion, we can construct an inverse completion $\struct {T', \oplus'}$ of $\struct {S', \oplus'}$, which is an isomorphic copy of $S$ under the mapping $\psi: S \to S'$.


This article, or a section of it, needs explaining.
In particular: $\psi: S \to S'$ has not been defined
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.



By the Embedding Theorem, there exists a semigroup $\struct {T, \circ}$ which contains $\struct {S, \circ}$, and an isomorphism $\Psi$ from $\struct {T, \circ}$ to $\struct {T', \oplus'}$ which extends $\psi$.


Thus $T$ is an inverse completion of $S$.

$\blacksquare$


Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $\S 20$: The Integers: Theorem $20.3$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.