User talk:Abcxyz/Sandbox/Dedekind Completions of Ordered Sets

Let me just check to be sure I'm understanding. Is this an alternative definition combined with an equivalence proof? --Dfeuer (talk) 04:47, 27 January 2013 (UTC)

Yes. Here's why I switched:

The original definition basically imitated Definition:Order Completion. As the uniqueness of $\phi$, if we take $\phi: T \to T'$ and $\phi {\restriction_S} = \operatorname{id}_S$, holds only if $\left({S, \preceq_S}\right)$ is already a complete lattice (or if I'm mistaken), I am inclined to think that the stated definition and its uniqueness proof are questionable. I searched for sources, but found none. However, I did find several sources that have the definition of the Dedekind completion as defined on this page (in the context of Riesz spaces, where condition $({2})$ follows easily), so I figured that it would be better to use this version as the definition. Furthermore, the only use of Characterization of Dedekind Completion (so far) is in the uniqueness proof (and a verification that it satisfies what one might think of as a "completion"). --abcxyz (talk) 16:56, 27 January 2013 (UTC)

I can't speak to the content (yet), but as I recall the uniqueness proof for order completion was based on a definition Lord_Farin replaced some time ago for the crime of not making any sense. I could, however, be misremembering. --Dfeuer (talk) 17:49, 27 January 2013 (UTC)