Talk:Distributive lattice

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Finally distributivity entails several other pleasant properties. For example, an element of a distributive lattice is meet-prime if and only if it is meet-irreducible, though the latter is in general a weaker property. By duality, the same is true for join-prime and join-irreducible elements.

Can someone provide a reference for this?--Malcohol 09:01, 16 October 2006 (UTC)

I have added a proof in Distributive lattice/Proofs. Ceroklis 21:14, 28 September 2007 (UTC)

External links

A link to a short document providing equivalent statements of distributivity was recently dismissed as spam. I do not understand the reasoning behind this. Surely you cannot argue that the document has NO merit. The material is directly relevant to the article. If you think it has LITTLE merit, then the "Related Links" section is precisely where it belongs. If you think it has MUCH merit, then it should have been incorporated into the article rather than deleted. Austinmohr (talk) 04:04, 20 March 2010 (UTC)

Hello, I removed the link for three reasons. First, because Wikipedia is not a collection of links. Although one or two external links may be appropriate, it's better to add information to the article, with references to reliable sources. External links sections can, over time, become swamped with links. Links should only be included if they add something to the article that would be missing if the article ever became a Featured Article; ie. it provides a unique resource that couldn't be provided by Wikipedia. I couldn't see anything when I followed this link that couldn't be added to the article. Please see Wikipedia:External links#Links normally to be avoided for more information. I would recommend you add useful information to the article, and reference it to reliable sources.

Secondly, the link is to a PDF file. Although that would be fine as a reference, it's best not have links in the EL section that require a reader to have an external application or plugin to read. This is also covered by Wikipedia:External links#Links normally to be avoided.

Thirdly, I was concerned by the fact that you appear to be linking to your own website, which is considered a conflict of interest, and could be seen as External link spamming.

Thankyou for bringing this up here, where others can weigh in if they wish to. If you have further concerns, feel free to ask. If you want other opinions on this (and they aren't forthcoming here) then you can also ask at Wikipedia:External links/Noticeboard. Regards, --BelovedFreak 08:52, 20 March 2010 (UTC)

And I meant to say that I have no particular knowledge of distributive lattices, so if others that do deem this to be a valuable link, then that's fine, but it should be someone unconnected to the website.--BelovedFreak 09:08, 20 March 2010 (UTC)

I am the author of Modular lattice. I saw your link here and the one at that article and decided not to bother doing anything about it at once, but I agree with Belovedfreak. Your notes provide very little insight. Moreover: There are free lecture notes available on the net, by distinguished lattice theorists and universal algebraists, that contain the same information, plus proofs, plus a lot more. It's absurd to link to a mere short list of equivalent properties. It might be useful for someone as a handy reference, but that's not sufficient reason to include the link. Hans Adler 11:48, 20 March 2010 (UTC)

Images

The left object in the picture captioned "Distributive lattice which contains N5..." is misleading in that it is not a Hasse diagram and so not a standard representation of a lattice; the line connecting b and c is extraneous (c is already clearly below b through the c-f-b connection). I would suggest removing that object and making separate objects displaying the pentagon with f removed, and the standard Hasse diagram with the connection between b and c erased.

The right-hand object is very confusing for some of the same reasons (the b-c and c-e connectors are extraneous), and the claim that M3 is a subset is incorrect. I believe the contributor is trying to demonstrate that by taking the full transitive closure of the Hasse diagram (almost) represented by the left image, you can find M3. A lattice is a set X with an order <, which can be represented by the Hasse diagrams; therefore a subset of the lattice is a subset of X with < restricted to the subset. Using the definition of subset, the claim that you can find M3 in a subset of this lattice is wrong. By taking the transitive closure of the diagram, you can find M3 as a *subdiagram* of the diagram (by forgetting some edges), but this is not the same as a subset of the lattice. JoelleJay (talk) 16:43, 10 May 2016 (UTC)

As far as I remember, I happend to find that picture at wikimedia commons, and tried to figure out its purpose from its appearance and its use in Polish wikipedia (without speaking any Polish); I didn't think too much it then. Also, you are right that the diagrams are not Hasse diagrams in a strict sense, as they both don't show transitive reductions of orderings.

That said, I still think they can be understood in a way such they make sense:

Each diagram represents two lattices, the first and second one being obtained as the transitive closure of the solid lines and of all lines, respectively. That is, in the left diagram the line c-b is redundant in the all-lines lattice (it doesn't hurt there, on the other hand) but is needed for the solid-only lattice. Similarly, the lines c-d, c-e, and b-a in the right diagram are redundant in the all-lines lattice, but needed in the solid-only lattice. (While I check the diagrams for writing this response, I see that admittedly some vertice names are hard to read, and the distinction solid/dashed is sometimes hard to recognize.)

All four (left/right diagram with solid-only/all lines) structures are in fact lattices, which can be checked by drawing them with irrelevant lines omitted. (Admittedly, the picture should be improved to ease that task - possibly the redundant lines could be in grey?) The solid-only lattices are obviously N5 and M3. The all-lines lattices are obviously ordered subsets of the solid-only lattices (viewed as ordered sets); in the right diagram, the vertice sets agree, but the ordering relations are one included the other.

So, I'd suggest to add a note that the pictures are Hasse diagrams not in the strict sense, but in a weak sense (allowing for redundant lines), and eventually to improve the picture (should better be svg, anyway). Would that be sufficient in your view, or did I miss something? - Jochen Burghardt (talk) 18:46, 10 May 2016 (UTC)

I suppose there is a question of what is meant by "contained as a subset." In the first case (on the left side), we have that the dark edges are giving an induced subposet, i.e., a subset of the vertices with induced order relation. In the second case (on the right side), we have that the dark edges are giving a non-induced subposet, i.e., we are taking both a subset of the vertices (well, all of them) and a subset of the relations. The first operation is natural; the second one is very not natural, and in particular no one would ever imagine that the second operation would preserve the lattice property. I think that for this reason the right half of the question is not very helpful. I do not have a problem with the non-Hasse diagram on the left side, this could be easily clarified in the caption. --JBL (talk) 19:45, 10 May 2016 (UTC)

Confused by intro

Will someone take the time to explain what is meant by the operations of "join" and "meet"? If articles about these operations exist a hyperlink will most likely suffice. 134.29.231.11 (talk) 21:49, 14 December 2011 (UTC) (resolved)

Representation theory may need a remark

In the representation theory section, it is stated that every distributive lattice is isomorphic to a lattice of sets, but the theorems cited for infinite lattices work for bounded lattices. It can be a little confusing; maybe we should add that every distributive lattice can be extended to a bounded one (by adding top and bottom if needed) without losing distributivity in the process. Jose Brox (talk) 12:04, 12 November 2017 (UTC)

Free Distributive Lattices Sequence

I wonder if omitting the 0 at the start of the second sequence of Dedekind numbers is fully justified. If we "disallow empty meets and joins", then the empty partial order is a valid option, which suggests its inclusion, but on the other hand lattices are required to have *some* top and bottom element, which makes the empty order not a (distributive) lattice. Since the sequence is preceded by a comment about there being two fewer elements, a comment should be made to justify the omission. M. Rogers, 29 March 2019

I think the omission of 0 is a typo; OEIS includes it, and mentions the free distributive lattice count. I checked the lattice definition in Gratzer.2003, and don't see why the empty partial order shouldn't be a lattice, and vacuously a distributive one. Therefore, I'll add the 0. - Jochen Burghardt (talk) 16:47, 29 March 2019 (UTC)

Figure on free distributive lattices

The lattices in the figure captioned "Free distributive lattices on zero, one, two, and three generators" include elements $0\neq \bigvee G$ and $1\neq \bigwedge G$. In other sources, I see the claim that in a free distributive lattice, $0=\bigvee G$ and $1=\bigwedge G$ holds. Which of those two is correct? — Preceding unsigned comment added by 131.174.142.107 (talk) 15:33, 24 July 2019 (UTC)

Yeah, those are technically not just free distributive lattices, but free distributive lattices with 0 and 1 adjoined (via coproduct). OEIS A007153 vs A000372. Balbes and Dwinger give the 18-element (not the 20-one) as example in their book on p. 90. On the other hand, as this page says "If empty joins and empty meets are disallowed, the resulting free distributive lattices have two fewer elements." I guess other authors consider empty meets and joins allowed without much discussion.

Non-equivalence of x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and its dual

@JayBeeEll: Being a non-native English speaker, I repeatedly looked up "inscrutable" in my dictionary, but I still don't understand what you mean. The N5 image show elements x,y,z such that x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) is wrong (as presented in the caption), but x ∨ (y ∧ z) = x ∨ 0 = x = 1 ∧ x = (x ∨ y) ∧ (x ∨ z) is right (as obtained by a similar computation). Therefore, while the universal closure of both equation is equivalent (as stated in the text section Definitions), the ternary relations defined by the unquantified equations are not (this is what the footnote tries to say). If you tell me what you find difficult to understand in the footnote, we can try to devise a btter sentence for it. No source is needed for the simple computation; we don't require a source for e.g. 2+3*4 = 14, either. - Jochen Burghardt (talk) 18:03, 28 October 2023 (UTC)

It is impossible to parse the assertion being made in the footnote: what is "the first equation", what is "the second equation", what is the context in which the assertion is supposed to apply, what is one supposed to check is or isn't true in N_5, what does it have to do with the definition of distributive? You don't need to tell me here the answers to these questions, I understand what the point is supposed to be; but that point is a textbooky aside about the importance of quantifiers, jammed into a totally inappropriate place and incomprehensible there as written. It could be expanded into a short paragraph that is decipherable by people who don't already know what it is trying to say (beginning, "In a lattice that is not distributive, it may be the case that ..." or something), but that would make it more inappropriate as an uncited footnote, because of WP:DUE and because Wikipedia is not a textbook. --JBL (talk) 18:40, 28 October 2023 (UTC)

I agree that adding an introductory sentence like your suggestion "In a lattice that is not distributive, it may be the case that ..." is a good idea. As for "first" and "second equation", I thought the meaning of this should be obvious when the footnote is placed after exactly two indented equations were shown in the text, but maybe this can be improved. Also, the forward reference to the N5 picture may need improvement. (By the way: I agree that the gallery should be dissolved into 2 thumbnail images; however, the M3 and the N5 image should be shown before the "subset vs. sublattice" image; your recent move destroyed that order.)

I meanwhile found 2 references:

• Brian A. Davey and Hilary Ann Priestley (1990). Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press. ISBN 0-521-36766-2. LCCN 89009753. Here: Remark 6.5.(4) on p.131.
• George Grätzer (2003). General Lattice Theory (2nd ed.). Basel: Birkhäuser. ISBN 3-7643-6996-5. Here: Exercise III.2.4 on p.189, resting on the definition of Ch. III.2 "Distributive, Standard, and Neutral Elements" (p.181-192; Thm.3 and 6 in this chapter originate from Grätzer).

I consider the footnote remark important even in an encyclopedia since the reader should be warned about a possible misunderstanding when applying the equivalence (∀x,y,z. x∧(y∨z) = (x∧y)∨(x∧z)) ⇔ (∀x,y,z. x∨(y∧z) = (x∨y)∧(x∨z)).

If you agree, we could devise textual improvements for the remark (including possibly a better place to put it); if not, this would be a waste of effort.

As an alternative, one might think of presenting more of Grätzer's section about "Distributive, Standard, and Neutral Elements" (in the long run); this might include the remark in question, or make it redundant and omittable. - Jochen Burghardt (talk) 08:01, 30 October 2023 (UTC)

Hi @Jochen, I am willing to put aside the objection to the textbookyness (though really "the reader should be warned" why should the reader be warned about this? it has nothing to do with distributive lattices per se) to discuss textual improvements; certainly I think including a reference would be a nice improvement to the article. I do not have access to these two sources but perhaps you can make a first suggestion based on their content?

I have fixed hopefully the issue with the images -- let me know what you think. --JBL (talk) 21:08, 1 November 2023 (UTC)

@JayBeeEll: The images look fine now; maybe their captions should be capitalized, similar to their common footer?

Triggered by your question "why", I came up with the suggestion to include a definition of "distributive element" (x is distributive if ∀y,z: x∨(y∧z)=(x∨y)∧(x∨z), Def.III.2.1(i) on p.181 of Gratzer.2003) and "dual distributive element" (the obvious dual definition, remark after Def.III.2.1 on same page), and relate the discussed footnote to Gratzer's exercise (III.2.4 on p.189) that both notions don't coincide in non-distributive lattices. (Now that I read it more carefully: while the exercise does imply the discussed footnote, is not a too obvious reference. However, Davey.Priestley.1990 is an appropriate reference, since it literally contains the footnote's formula.)

Then we could create a redirect distributive element to these explanations. My answer to your "why" question would then be: to explain the difference between "distributive lattice" and "distributive element" (and "distributive triple" (x,y,z)), and the equivalence and non-equivalence (and again non-equivalence) of their dual versions, respectively. Beyond that, the redirect "distributive element" might eventually be extended to an own stub, sourced (at least) by Gratzer's Ch.III.2.

You can see from the grey text that I had to revise my original suggestion; I feel, however, that it might still be worthwhile to think along these lines.

I could send you a scan of the pages in question, but they are too many to be upload here, even temporarily, I'm afraid. - Jochen Burghardt (talk) 09:46, 2 November 2023 (UTC)

I'm skeptical that this article makes sense as a home for a definition of distributive element in a (not necessarily distributive) lattice, for the same reason I wouldn't frame the invertibility axiom of groups in terms of group-like elements in a monoid: it's digressive from the topic of this article. The article Distributivity (order theory) would be a much more natural home for that, in my opinion.

And, sure, if it's not a burden, I would be happy to see the pages. --JBL (talk) 20:19, 5 November 2023 (UTC)

I just added a new section to distributive lattice, and have to leave Wikipedia for today. Moving to Distributivity (order theory) seems indeed a good idea, I'll look at this tomorrow. - Where do I have to look in order to send an E-Mail to you? I knew such things once, but I've forgotten them. - Jochen Burghardt (talk) 21:40, 5 November 2023 (UTC)

 Done I performed the move. - Jochen Burghardt (talk) 09:27, 6 November 2023 (UTC)

Thanks, that's a nicely written section and I think the placement makes a lot of sense. If you visit my user page, there should be a link that says "E-mail this user" somewhere in one of the menus (in the view I use it's on the left sidebar). --JBL (talk) 00:58, 10 November 2023 (UTC)