422:, 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. --
331:. 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.
686:
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
84:
670:⨠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. -
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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.
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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.)
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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
464:
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
357:
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
201:
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
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Triggered by your question "why", I came up with the suggestion to include a definition of "distributive element" (x is distributive if ây,z: xâ¨(yâ§z)=(xâ¨y)â§(xâ¨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
391:
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
652:
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
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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
685:
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
940:
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
484:
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
353:
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
166:
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
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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. -
1071:
The reference for Voigt redirects to his wikipedia page, not the paper where he proved it. Dedekinds reference is for a biography of Peirce, which has nothing to do with
Dedekind. Can someone fix these references and add the real sources for where they are from? I cannot find them.
399:
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? -
720:", 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.
904:(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.)
842:
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. -
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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.
276:
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
245:
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
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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
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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.--
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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.
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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.
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224:. 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
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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).
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228:; ie. it provides a unique resource that couldn't be provided by Knowledge. I couldn't see anything when I followed this link that couldn't be added to the article. Please see
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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.
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I consider the footnote remark important even in an encyclopedia since the reader should be warned about a possible misunderstanding when applying the equivalence (âx,y,z.
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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.
444:
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.
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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. -
202:
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.
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to these explanations. My answer to your "why" question would then be: to explain the difference between "distributive lattice" and "distributive element"
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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
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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.
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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.
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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.
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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
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1012:, 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). --
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722:(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
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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. -
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The lattices in the figure captioned "Free distributive lattices on zero, one, two, and three generators" include elements
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Thirdly, I was concerned by the fact that you appear to be linking to your own website, which is considered a
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for more information. I would recommend you add useful information to the article, and reference it to
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Thanks, that's a nicely written section and I think the placement makes a lot of sense. If you visit
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The images look fine now; maybe their captions should be capitalized, similar to their common footer?
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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should the reader be warned about this? it has nothing to do with distributive lattices per se)
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References for independence proofs of distributivity are bad for Dedekind and Voigt
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I have fixed hopefully the issue with the images -- let me know what you think. --
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That said, I still think they can be understood in a way such they make sense:
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the "subset vs. sublattice" image; your recent move destroyed that order.)
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And, sure, if it's not a burden, I would be happy to see the pages. --
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I agree that adding an introductory sentence like your suggestion "
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In a lattice that is not distributive, it may be the case that ...
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738:. Cambridge Mathematical Textbooks. Cambridge University Press.
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Non-equivalence of x ⧠(y ⨠z) = (x ⧠y) ⨠(x ⧠z) and its dual
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I am willing to put aside the objection to the textbookyness
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Hello, I removed the link for three reasons. First, because
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Korselt's non-distributive lattice example is a variant of
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would be a much more natural home for that, in my opinion.
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Knowledge:External links#Links normally to be avoided
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Knowledge:External links#Links normally to be avoided
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734:Brian A. Davey and Hilary Ann Priestley (1990).
867:(though really "the reader should be warned"
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694:and because Knowledge is not a textbook. --
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171:Can someone provide a reference for this?--
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480:Free Distributive Lattices Sequence
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1096:"Bemerkung zur Algebra der Logik"
565:{\displaystyle 1\neq \bigwedge G}
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764:(2nd ed.). Basel: Birkhäuser.
756:Here: Remark 6.5.(4) on p.131.
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623:{\displaystyle 1=\bigwedge G}
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454:21:49, 14 December 2011 (UTC)
109:and see a list of open tasks.
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183:Distributive lattice/Proofs
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