Definition talk:Complete Lattice/Definition 2

Merge?
I vote we turn this into a disambiguation page, probably synonymous with Definition:Complete Ordering. There are:


 * Dedekind completeness
 * Bounded completeness (Dedekind both directions)
 * dcpo (often called just cpo)
 * Complete lattice
 * Wikipedia also mentions omega-complete partial order, but I haven't encountered that elsewhere yet, personally, which of course doesn't mean it's not real.

So ... might be good to organize these things. --Dfeuer (talk) 00:53, 10 January 2013 (UTC)


 * Sounds good to me.
 * By the way, bounded completeness is not the same as Dedekind completeness in both directions; Dedekind completeness is a self-dual property. --abcxyz (talk) 01:28, 10 January 2013 (UTC)


 * Headdesk. Sorry 'bout that... I got very little sleep and have a headache. --Dfeuer (talk) 01:52, 10 January 2013 (UTC)


 * First of all, the disambig you propose should probably reside at "Complete Ordered Set". $\omega$-completeness is that every chain has an upper bound (or lower, whichever matches the categorical notion). Awodey speaks of it but I haven't covered that yet. --Lord_Farin (talk) 08:48, 10 January 2013 (UTC)
 * Oh, and I forgot to say I'm supporting the idea. --Lord_Farin (talk) 08:49, 10 January 2013 (UTC)

Disambig or not
I have revisited this suite of pages.

I actually don't recommend Definition:Complete Ordered Set to be a disambig, for the reasons:
 * a) we already have Definition:Complete, which should suffice (and has now been expanded as appropriate in our house style to make it more useful)
 * b) as a complete ordered set is a well-defined concept in its own right, it merits its own page, and it is not clear how the disambig would work here
 * c) we can include the different types of "completely ordered" in the "also see" section, as I have done.

I think what I've put together should work as a general strategy. --prime mover (talk) 03:18, 14 February 2018 (EST)


 * Seconded. &mdash; Lord_Farin (talk) 13:39, 14 February 2018 (EST)

Merge or not
I say no to the merge, as an ordered set is not (necessarily) a lattice. So while a "complete lattice" is an instance of a "complete ordered set", if anything the connection should be at the "Lattice" end: a "Complete Lattice" is a "lattice" which is also a "complete ordered set". --prime mover (talk) 03:22, 14 February 2018 (EST)


 * Actually, the definition of lattice is subsumed in the very general condition for "complete ordered set" (if you check Definition:Lattice). Therefore a merger actually does make sense. &mdash; Lord_Farin (talk) 13:39, 14 February 2018 (EST)


 * Yes I know what a lattice is. The question is, since when did we take the general and merge it into the particular? Might as well merge the definition of "topological space" and merge it into the definition for a real number line, because a real number line is a topological space. --prime mover (talk) 14:01, 14 February 2018 (EST)


 * Oh, come on, debate using arguments for a change. As I said, the concepts of "complete lattice" and "complete ordered set" coincide completely because the condition added to an ordered set to make it complete subsumes the added condition to make it a lattice. Therefore we have a linear extension OrdSet -> Lattice -> CompleteOrdSet / CompleteOrdLattice and the distinction if we arrive at the strongest form via the intermediate step of a lattice, or arrive there directly, is immaterial. Like insisting that the direct definition of group and that as a monoid whose operation is inverse-complete are completely alien to one another. &mdash; Lord_Farin (talk) 14:46, 14 February 2018 (EST)


 * So what we need is a page saying that a "completely ordered set" and a "complete lattice" are the same thing. That is not obvious and I admit (through lack of thinking about it hard) that I still can't immediately see why.


 * There's a case, then, for implementing it as a double definition with equivalence page and redirect. I'll let someone else do it, as my brain is somewhere else at the moment. --prime mover (talk) 15:36, 14 February 2018 (EST)


 * That would be the best option indeed. &mdash; Lord_Farin (talk) 17:13, 14 February 2018 (EST)

Finally got my brain in gear for this one. Of course a complete ordered set is exactly the same as a complete lattice, because a complete ordered set is a subset of itself, and what it is saying is that a "complete ordered set" is an ordered set such that every subset is a lattice. Hence it is a lattice itself. I will do what needs to be done. --prime mover (talk) 02:51, 7 April 2020 (EDT)