Talk:Complete lattice

Why does Cardboard give this page, which appears to be called "Complete lattice"? 143.252.80.124 12:34, 28 July 2005 (UTC)[reply]

Representation

The "Representation" section claims that every complete lattice is isomorphic to a lattice of sets. M_3 and N_5 are examples of complete lattices that, being non-distributive, are not isomorphic to lattices of sets. Would someone in the know mind checking up on this? —Preceding unsigned comment added by 82.19.52.143 (talk) 21:19, 11 November 2010 (UTC)[reply]

Complete sublattice

The article should include a definition of complete sublattice (please cite source). Although I'm fairly sure I know the definition, I can't find a source right now.--Malcohol 08:57, 5 July 2006 (UTC) Actually, I found one in that reference at the bottom of the page. I've added its definition. --Malcohol 09:39, 5 July 2006 (UTC)[reply]

Counterexamples?

Could somebody add some counterexamples please? Examples that suit the definition are not enough to understand a concept.. Thanks, Alex —Preceding unsigned comment added by 141.76.179.192 (talk) 12:59, 8 May 2011 (UTC)[reply]

Examples

As the comment above states, there are plenty of examples of a complete lattice (most of them a variation of set inclusion). In fact it would seem that all "commonly encountered" lattices are complete. Rather than having 14 different examples of a complete lattice and no example of a non-complete lattice, it would be much more useful to have 2-3 examples of a complete lattice and 2-3 examples of a non-complete lattice. --84.245.121.46 (talk) 21:28, 17 December 2020 (UTC)[reply]

I have added 2 non-examples of a complete lattice. I agree that we should remove some examples of complete lattices and add a couple more non-examples. I am not sure which ones would be suitable candidates to remove. Spamakin (talk) 15:54, 14 August 2024 (UTC)[reply]

I think the following would be fine to remove:

unit interval and extended real line
either the subgroup lattice or the submodule lattice
convex subsets of real / complex vector space
lattice of all topologies on a set
lattice of sub-multisets
lattice of all equivalence relations
lattice of self-adjoint projections

Would love to hear thoughts on this before killing those items. I will probably kill these items in a week or so if no other feedback occurs. Spamakin (talk) 22:22, 14 August 2024 (UTC)[reply]

As mentioned above, these examples have been removed. See https://en.wikipedia.org/w/index.php?title=Complete_lattice&diff=prev&oldid=1242237083 for the diff Spamakin (talk) 19:00, 25 August 2024 (UTC)[reply]

Incorrect definition of conditional completeness

Hello. I'm completely () new here so sorry if I've done anything wrong!

I think there are some problems with the definitions of "conditional completeness" in this article. In particular I believe the following two bits are not right:

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a conditionally complete lattice.

A lattice is said to be "conditionally complete" if it satisfies either or both of the following properties:^[1]

Any subset bounded above has a least upper bound

Any subset bounded below has a greatest lower bound
A lattice which satisfies both of these properties is known as a complete lattice, as well as a conditionally complete lattice.^[2]

These amount to three different definitions! I think very first definition about only having suprema or infima is incorrect. In fact, either one implies both, which the semilattices bit mentions. This is a pretty standard result that's proved eg here.

Then in the second part, it first defines "conditionally complete" as being at least one of the bullet points, then says both bullet points together characterises completeness, then says that's also called conditional completeness. There's a bit to unpack here. The last definition of conditional completeness does agree with the given source, but I still don't think it's right.

I think that the bullet points should be changed to also require the sets are non-empty, and a lattice is conditionally complete iff it satisfies both of those new bullet points. This is different to the PDF notes used as a source, which I believe are wrong (since without a non-emptiness requirement, any such non-empty lattice automatically has a maximum and a minimum element - the inf and sup of the empty set, respectively, so it's automatically complete). There should be no mention of "complete lattice" in the context of these bullet points.

This is the definition given for instance here. I appreciate that I haven't given a proper source, but I'm struggling to find one. The mathlib source is I think quite convincing even if it's not technically acceptable as a source, since it and its consequences have been computer-verified.

None of the three definitions currently in the article are equivalent to this definition, as for instance the set of real numbers $\mathbb {R}$ with its usual order ought to be conditionally complete, but is not according to those definitions.

I'm a bit hesitant to wade straight in and start editing because I'm so new - I'm not sure I have a complete handle on all the style conventions and formatting technology. Also I don't have a good source on hand! Sorry if this violates "be bold"...

Goedel Gang (talk) 17:07, 7 July 2022 (UTC) Goedel Gang (talk) 17:07, 7 July 2022 (UTC)[reply]

I agree with you, the first definition seems incorrect by omitting the "bounded" restriction. If we remove the bounded restriction, then "conditionally complete lattice" would be a synonym for "complete lattice".

I disagree on a point of your point about emptiness. Having a minimum and a maximum does not imply being complete. You cannot be unbounded and complete, but you can be bounded and incomplete.

And I would also agree on removing "a complete lattice, as well as".

Now, I am french and not used to the exact terminology, but at least making these choices seems to make a coherent set of definitions, unlike what is currently done.

Sedrikov (talk) Sedrikov (talk) 15:26, 7 February 2024 (UTC)[reply]

References

^ Baker, Kirby (2010). "Complete Lattices" (PDF). UCLA Department of Mathematics. Retrieved 8 June 2022.
^ Kaplansky, Irving (1972). Set Theory and Metric Spaces (2nd ed.). New York City: AMS Chelsea Publishing. p. 14. ISBN 9780821826942.

[1] Baker, Kirby (2010). "Complete Lattices" (PDF). UCLA Department of Mathematics. Retrieved 8 June 2022.

[2] Kaplansky, Irving (1972). Set Theory and Metric Spaces (2nd ed.). New York City: AMS Chelsea Publishing. p. 14. ISBN 9780821826942.

[1]

[2]

Representation

Category

Complete sublattice

Counterexamples?

Examples

Incorrect definition of conditional completeness