Talk:Lattice (order)

message: There is a mistake in the characterization of atomistic property, as now, it contradicts transitivity of the lattice order.  — Preceding unsigned comment added by 210.74.131.204 (talk) 09:13, 9 July 2018 (UTC)
thread_title:  Atomistic property 

message: The list of examples of complete lattices contains this item:
*The non-negative integers, ordered by divisibility. The supremum is given by the least common multiple and the infimum by the greatest common divisor.
What is the least common multiple of an infinite set of non-negative integers?  How is this lattice then complete?  It's not even bounded, is it?  Am I missing something?  Also, the issue of completeness aside, should that be positive integers?
Or have I just answered my own question?  Zero could be considered the least common multiple of an infinite set, and a bound on the lattice.  But then least here doesn't refer to the usual ordering of the integers, or zero, if considered a multiple of everything, would be the least common multiple of every set.  Perhaps a little more explanation is in order Josh Cherry 23:26, 8 Jul 2004 (UTC)
replies: text: Yes, that is basically correct: "least" and "greatest" refer to the given order of divisibility, not to the normal total order of integers. 0 is then the greatest element: it is divided by any number. 1 is least: it divides any number. Infinite sets have only 0 as an upper bound, while they may have numbers other than 1 as a lower bound (like 2 in the case of all even numbers). I will add some remarks... --Markus Krötzsch 14:57, 15 Jul 2004 (UTC)
replies: text: Cool, thanks.  I added a sentence to that section.  Please verify that it's correct. Josh Cherry 22:15, 15 Jul 2004 (UTC)
replies: text: Yep, that's true. --Markus Krötzsch 14:03, 20 Jul 2004 (UTC)
thread_title:  Non-negative integers 

message: I'm surprised this article doesn't talk about top and bottom anywhere, although it does use the terms (with no defining link). Are these described elsewhere or this something that should be added? Deco 18:52, 16 December 2005 (UTC)
replies: text: Those are not technical terms.  I guess someone used those terms to try to be friendlier to laymen.  The definition of bounded lattice can be found later down the article: a bounded lattice is a lattice with a least and greatest element.  The words "top" and "bottom" are more or less meaningless.  Maybe they make sense when you have the Hasse diagram of the lattice in front of you, vertically aligned. -lethe talk 19:12, 16 December 2005 (UTC)
replies: text: Well, I've encounter them a lot in classes and such. They seem to be conventional names for the least and greatest element of a bounded lattice, in the same way that p is a conventional name for a positive prime integer. I think they're frequently used enough to justify mentioning, at least in passing. Deco 20:37, 16 December 2005 (UTC)
replies: text: I've never heard that before in my life, though surely there are a lot of things I've never heard of.  But can you tell me a textbook that uses that terminology, if you know one off-hand?  I'd like to see it for myself.  -lethe talk 21:08, 16 December 2005 (UTC)
replies: text: Davey and Priestley, Introduction to Lattices and Order () uses top and bottom. As far as I now they are in fact technical terms, at least in domain theory and its applications to denotational semantics. — Tobias Bergemann 22:48, 16 December 2005 (UTC)

text: Also see greatest element. — Tobias Bergemann 22:58, 16 December 2005 (UTC)
replies: text: Ah, that's why they're familiar to me. I encountered them in programming language semantics. I also encountered them studying compiler optimizations like sparse conditional constant propagation. Seems CS people like them. Deco 23:29, 16 December 2005 (UTC)

text: I think it's even worse. As far as I know (I could be wrong, though), the lattice theories in physics have nothing at all to do with lattice (order), but are somehow related to lattice (group). Which is something entirely different. --Hans Adler (talk) 10:52, 18 April 2008 (UTC)
thread_title:  Top and bottom 

message: If no one objects within a couple of days, I'll add a section on dual lattices (reversing the partial order, or, equivalently, substituting meet by join and vice versa).  This duality is pretty obvious to any lattice theoretician, of course, but not necessarily so for the readers.  It is interesting in itself, and I'd like to refer to it in order to clarify why e.g. meet-semilattices and join-semilattices coincide as abstract algebraic structures.  I also found no other article dealing with this particular duality.
(Actually, it might be possible to outline connections to other kinds of duality.  E.g., recall that if (L,+,\cap) is the (modular) lattice of all subspaces of a finite-dimensional vector space V over a field k, then the dual lattice may be canonically identified with the lattice of subspaces to the dual vector space V^* =  Hom(V,k), by letting W ∈L correspond to \{ f \in V^* : f(W)=0 \}.  However, I suspect that this would lead too far for the Lattice (order) article.) JoergenB 18:42, 21 September 2006 (UTC)
thread_title:  Dual lattices 

message: Partially ordered set, shown on picture, is not a lattice! For example, elements 1234 and 12/2/4 have no infimum. Dims 05:20, 16 January 2007 (UTC)
replies: text: Are you sure? If you mean 1234 and 12/3/4 then their infimum is 12/3/4 and their supremum is 1234 as 1234 > 12/3/4. --- Richard CHSTalk 22:49, 18 January 2007 (UTC)
thread_title:  It is not lattice on picture 

message: Is a lattice necessarily acyclic?  If so, that is a key fact that would be helpful to mention explicitly.  Based on reading the definitions of supremum and infimum, I would assume that a lattice is acyclic, but not being well versed in this area of mathematics I would have to spend much more time reading and thinking it through to convince myself with certainty, so it would be nice to simply hear it from an expert.  Of course, implicit here is the fact that I'm thinking of a lattice as a kind of graph, which I hope is a reasonable way to think about it, and one which I also think would be helpful to mention.  Thanks.  -- DBooth 21:40, 22 January 2007 (UTC)
thread_title:  Acyclic? 

message: I'm thinking of moving the section called "free lattice" to its own article; would there be objections to that? linas 16:40, 6 April 2007 (UTC)
replies: text: I'm gonna do it, unless someone says no...linas 04:34, 19 April 2007 (UTC)
replies: text: Done. See Free lattice. linas 14:18, 8 June 2007 (UTC)
thread_title:  Free lattice 

message: I'm not a mathematician at all, but I'm dealing with a physical model that can be viewed as a poset - yet I'm not certain at all if it is a lattice.
The graph structures produced are always most disordered and complex, and it's not easy to grasp if inf{} and sup{} do exist always.
I was wondering if some rule of thumb exists, to judge the lattic-ity of a poset.
E.g. if a poset is 1) connected and 2) does not exhibit a "butterfly" pattern (fig.2.1(ii) in "Introduction to Lattices and Order") is it a lattice?
references:
G. Bertotti, P. Bortolotti, A. Magni, V. Basso Topological and energetic aspects of the random-field Ising model J.App.Phys.101(2007)09D508
P.Bortolotti, A.Magni, V.Basso, G.Bertotti Dynamical-system theory approach to the study of metastable states in the random field Ising model  J.Magn.Magn.Mat. 310(2007)1543-1545
thanks for any hint, 193.204.114.65 10:07, 7 August 2007 (UTC)
thread_title:  Conditions for a poset to be a Lattice 

message: I want examples of (infinite) partially ordered sets which are not lattices and even not semilattices.
porton (talk) 18:10, 28 November 2007 (UTC)
replies: text: The purpose of this page is to discuss improvements to the article.  This question belongs on Wikipedia:Reference desk/Mathematics instead. Michael Slone (talk) 20:26, 28 November 2007 (UTC)
replies: text: But giving examples of posets that aren't lattices is an improvement to the article.  I gave a few such counterexamples just now, all finite but they're easily made infinite by disjointly adding any infinite poset.  --Vaughan Pratt (talk) 21:29, 2 December 2009 (UTC)

text: That's Lattice multiplication and it has very little to do with the theory of partially ordered sets. —David Eppstein (talk) 01:25, 14 February 2008 (UTC)
thread_title:  Examples of posets which are not lattices 

message: "In mathematics, a lattice is a partially ordered set (or poset)": Ambiguous? I am guessing that it means "In mathematics, a lattice is a partially ordered set (a poset)" but I have not the expert knowledge of the topic necessary to be certain that is in fact what it means, rather than what it well might, taken as English, mean, to wit that it is EITHER a partially ordered set (see explanatory link if not sure what that means) OR a poset (oops no clickability of this term so no idea what the heck does it mean).
My immediate impulse as a Wikipedia Proofreader was to edit it to change 'or' to 'a' in order to make it less susceptible to being interpreted the wrong way, aka IMO to make it 'less ambiguous' aka 'more clear'. But what if the author already did this soul-searching, determined with mathematical rigour that 'a' would be incorrect, that 'or' is correct, yet somehow managed to neglect to make 'poset' an active hyperlink? I guess the reader should first search for themselves to discover whether there is or is not an entry in the Wikipedia explaining what a poset is? Maybe I should do that search and if such a page exists make 'poset' an active link?
I dunno, would using 'a' instead of 'or' be such an awful thing that all this extra work is preferable to using the term 'a' instead of the term 'or' ???
Knotwork (talk) 09:20, 18 April 2008 (UTC)
replies: text: Fixed. I have also moved your comment down and added a title. It's better to use the little "+" at the top of the talk page to start a new topic. --Hans Adler (talk) 10:46, 18 April 2008 (UTC)
thread_title: Partially ordered set or poset

message: Someone has proposed to merge this article with partially ordered set. While I am a great merger myself, I strongly disagree in this case. Lattices are not just a special case of partial orders in which any two elements have a least upper bound and a greatest lower bound. Even though the objects are the same in the finite case, we have the following crucial differences that are hard to handle in a single article:
* Infinite partial orders are normally not lattices at all.
* Even in the finite case, a homomorphism between partial orders (order-preserving map) is not normally a lattice homomorphism (sup- and inf-preserving map).
* Partial orders can (in general) only be understood as relational structures. Lattices, on the other hand, also have an alternative interpretation as algebraic structures.
General order theory on one hand and lattice theory on the other hand are very different subjects. While each comes up at the beginning of a course that teaches the other, it makes no sense to cram them into a single article. Both articles are already big enough to stand on their own, each with potential for more, and there would be no synergy effects from merging, only confusion. Hans Adler 15:42, 25 April 2010 (UTC)
replies: text: I am in complete agreement with you. I'm going to be WP:BOLD and remove the merge tags. —David Eppstein (talk) 16:12, 25 April 2010 (UTC)

text: Indeed. Paul August ☎ 16:42, 25 April 2010 (UTC)
thread_title:  Merger proposal 

message: "... in which any two elements have a unique supremum (the elements' least upper bound; called their join) and an infimum (greatest lower bound; called their meet)..."
I am under the impression that any two elements of a poset either do or don't have a supremum (join). If they do have a supremum, then the supremum is provably unique. It is therefore tautologous to say that any two elements are required to have a "unique" supremum. The current definition seems to acknowledge this by omitting the requirement that the infimum must be unique (although the wording is indeed slightly ambivalent in this regard). 
Bottom line: I suggest dropping the word "unique". Am I missing something? CarlosChio (talk) 12:00, 22 November 2010 (UTC)
replies: text: In the poset in which a,b are both above c,d, then a and b are both minimal upper bounds of c and d, but there is not one unique minimal upper bound. The "unique" is to emphasize that this sort of situation isn't good enough. —David Eppstein (talk) 16:46, 22 November 2010 (UTC)
thread_title:  Definition seems to contain a redundancy 

message: The article says: "The algebraic interpretation of lattices plays an important role in universal algebra and in economics of persuasion." It is unclear what "economics of persuasion" refers to -- there should be a reference, or this should be deleted. 
Krifka (talk) 05:03, 2 August 2011 (UTC)
replies: text: This was added without explanation by an anonymous editor on 4 April. It doesn't seem relevant to me, so I have simply removed it. Hans Adler 16:56, 2 August 2011 (UTC)
thread_title:  Economics of persuasion? 

message: I edited the article that "defined" the join and meet of the empty set, and justified this "convention."  It is neither a definition nor a convention, because it necessarily follows from the definition of join and meet, and the vacuous truth that every element must be a lower bound and upper bound of the empty set.
I thus removed the words "defined" and "convention," and, with hope, added sufficient text to explain how meet and join are derived. 149.125.250.184 (talk) 16:30, 27 April 2012 (UTC)
thread_title:  Join and meet of the empty set is not a convention 

message: This article suffers from the same problem that affects many mathematics articles.   The article is written in the abstract, which is fine for many mathematicians who think very much in the abstract, but not sufficient for an encyclopaedia who's purpose is to make knowledge accessible to a wider audience.   Specifically the article lacks an explanation of the purpose and relevance of Lattices in the real (non abstract) world and lacks examples of its application to problems that can be easily visualised by an audience with limited mathematical training.   In my case, it is difficult for me to understand whether this branch of mathematics would be useful to me or not.   This is an important question as it determines whether I ought to invest time in learning it (or spend the time on something else).   
Correcting the omission would help to bridge the gulf that exists between mathematicians and the rest of the population, to the benefit of both sides.  FreeFlow99 (talk) 09:07, 25 August 2013 (UTC)
thread_title:  Explanation 

message: I'm not sure that Pic.7's description is correct. Why would f be a common upper bound for b and c, but a not be a common lower bound?
What defines a common bound? Can someone with better expertise on this subject verify this diagram and its description? — Dbechrd (talk) 17:25, 22 July 2014 (UTC)
replies: text: I think you are misinterpreting the caption. a is a common lower bound. The reason this is not a lattice is different: it's because b and c have no lowest common upper bound. Among the three upper bounds d, e, and f, none is uniquely lowest. —David Eppstein (talk) 17:50, 22 July 2014 (UTC)
thread_title:  Counter-examples 

message: Every lattice has an empty sublattice - according to the general definition of substructure in universal algebra, or subobject in category theory. It is not natural to exclude it. -- Martin B.  --92.228.253.9 (talk) 23:06, 2 August 2015 (UTC)
thread_title:  Sublattices 

message: I did not understand the following sentence:
It follows by an induction argument that every non-empty finite subset of a lattice has a join and a meet.
How can this be true? Consider poset on [4] with relations 1<2, 1<3, 2<4, 3<4, 1<4 (which is a lattice). The subset {2,3} does not have a join or a meet.  — Preceding unsigned comment added by 62.46.197.84 (talk) 16:55, 19 November 2016 (UTC)
replies: text: I think the sentence is ambiguous. When understood as "each non-empty finite subset of a lattice is again a lattice, having both a join and a meet operation", it is wrong, as your example shows. However, when understood as "each non-empty finite subset of a lattice has a greatest lower bound as well as a smallest upper bound", it is true, and in fact follows by induction on the set's cardinality. - Jochen Burghardt (talk) 18:27, 19 November 2016 (UTC)
thread_title:  Subsets of a lattice 

message: imho the statement "A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet." is wrong (see eg. Pic. 7 for a bounded poset). I guess a correct (and simpler) statement would be "A lattice is bounded if and only if the empty set of elements has a join and a meet.". Other (nonempty) finite sets are a matter of induction, and mentioned elsewhere. Am I right?  — Preceding unsigned comment added by 187.3.207.124 (talk) 11:49, 24 March 2019 (UTC)
replies: text: With these operations, it is not true that a1 ≤ a2 nor that b1 ≤ b2, because (for these operations), the comparison operation is by divisibility, not numerical value. —David Eppstein (talk) 07:28, 15 May 2020 (UTC)
thread_title:  Mistake about bounded lattices in section 1? 

message: "This definition makes ∨ and ∧ binary operations. Both operations are monotone with respect to the order: a1 ≤ a2 and b1 ≤ b2 implies that a1 ∨ b1 ≤ a2 ∨ b2 and a1 ∧ b1 ≤ a2 ∧ b2."
If we take a1 = 5, b1 = 5, a2 = 6 and b2 = 8 and use the LGM, GCD as the binary operations, then the assertion above doesn't appear to hold.  a1 <= a2 and b1 <= b2, but a1 ^ b1 is 5 while a2 ^ b2 is 2, so a1 ∧ b1 ≤ a2 ∧ b2 doesn't hold.  
----  — Preceding unsigned comment added by Sophianto (talk • contribs) 06:48, 15 May 2020 (UTC)
replies: text: With these operations, it is not true that a1 ≤ a2 nor that b1 ≤ b2, because (for these operations), the comparison operation is by divisibility, not numerical value. —David Eppstein (talk) 07:28, 15 May 2020 (UTC)
thread_title:  Counter example ? 

message: The article states that "one can think of join and meet as binary operations that are defined on non-empty finite sets, rather than on elements". I interpret this as operations \bigvee and \bigwedge on sets as defined earlier in the article, e.g. a \vee b \vee c is rewritten to \bigvee \{a,b,c\}, but then the operations \bigvee and \bigwedge are not binary. Is my understanding incorrect, or is the word binary incorrect? — Preceding unsigned comment added by 131.174.142.107 (talk • contribs) 
:I think you're right. If you remove "binary" I would also change it to "rather than on pairs of elements". —David Eppstein (talk) 17:51, 23 July 2019 (UTC)
thread_title:  Join and meet on finite sets 

message: I'm confused what the article is claiming (and that's not clarified in the linked article on Distributive Lattice).
The narrative to Pic 11 says that the N5 lattice violates one distributivity equation, but satisfies its dual.
The main body gives those equations (but with different variable names and in reverse order) then says
> A lattice that satisfies the first or, equivalently (as it turns out), the second axiom, ...
If N5 satisfies the second but not the first, those equations can't be equivalent?
Does it mean the first equation implies the second, but not v.v.? AntC2 (talk) 23:37, 4 October 2019 (UTC)
thread_title:  Distributivity (section 8.3) 

message: which is at Top element (which oddly calls itself "Greatest element and least element" which it claims was "(Redirected from Top element)" but the URL is as given above, ending in /Top_element - wut??).
Any reason not to, because it would have been a good to have known about this WRT my earlier question here . If there is already such a link I can't find it. 79.75.122.175 (talk) 20:32, 7 August 2021 (UTC)
replies: text: You may want to read Wikipedia:Redirect, that should clarify a thing or two for you. As regards the link you want added, please note that both greatest element and least element already redirect to greatest element and least element. Paradoctor (talk) 20:55, 7 August 2021 (UTC)
replies: text:  Will check the redirect info, thanks. Didn't find the links you gave because I was looking for the whole title rather than just parts ie. didn't look just for 'greatest element'. Much obliged. 79.75.122.175 (talk) 21:16, 7 August 2021 (UTC)
thread_title:  Should link to article "Greatest element and least element" 

message: I think @Timeroot is right in that the current article gives WP:UNDUE weight to distributive lattices. The examples in the lead are distributive, and pretty much all the other examples are too; the only non-distributive examples are (confusingly enough) found in the distributive lattices section. In actual mathematics distributive lattices are common but so are non-distributive lattices; the article should not be solely about one or the other, and particularly considering that Distributive lattice is a page, I would say this article should lean towards discussing non-distributive examples. Mathnerd314159 (talk) 21:20, 28 January 2024 (UTC)
replies: text: Hi, thanks. @Jochen Burghardt, is the only objection that it messed up the numbering of the pictures? I can fix that. Or is there some other objection to the choice of pictures (too many, too 'busy' as pictures?). Let me know. Timeroot (talk) 16:55, 29 January 2024 (UTC)
replies: text: Having messed up the numbers (and having duplicated the N5 and M3 picture) is my only strong objection. For your A4*C2 image, I'd prefer a less complex version, e.g. similar to :File:Octahedral subgroup tree tidy.svg, where in interior details of vertices are omitted, but I wouldn't insist on that. While I understand Pic.1-5, I understand none of your new non-distributive examples, but this may be a matter of my (missing) foreknowledge. Nevertheless, if it was possible to add some easier examples, I'd appreciate that, and I guess, many readers would, too. - Jochen Burghardt (talk) 19:00, 29 January 2024 (UTC)
thread_title:  Non-distributive lattices