Talk:Order theory

Initial comments
"Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering.": The term 'mathematical ordering' wants to be a link, why? What is it intended that the page linked to will say? My immediate thought was how would a mathematical ordering differ from a mathematical sequencing, thus how do the terms or concepts sequence and order differ, which led me to think of terms such as ordinality and to wonder what order of complexity the concepts of sequentiality and being-in-order might have in this branch of mathematics? Has the author checked whether there is already a page explaining the concept he or she intends, but that page has a label not quite exactly the same as the label the author tried to use but for some reason has not yet gotten around to editting into a page?

Why is there not a page named 'mathematical ordering' if the term is so important to this important page? Why did the author not try different but similar constructions, like maybe 'mathematical order' or 'order (mathematics)' or 'order (order theory)' ?

I even wonder whether the term 'mathematical ordering' should maybe best be made into a shortcut pointing to the order theory page itself, as where else would I be likely to find a more comprehensive explanation of the concept of mathematical ordering than in the actual theory of order (which maybe itself should be labelled 'Order theory (mathematics)' for greater clarity and precision, maybe even rigour (lets not confuse it with 'order theory (waitressing)', 'order theory (heirarchical organisation administration)' or other concepts of order or ordering ;))

Maybe I should check to see if there are any pages on 'mathematical sequencing', 'sequence (mathematics)', 'sequence (order theory)' or the like? Or is order, in order theory, not at all similar to or related to sequence (is there such a thing as sequence theory???)

Etc etc etc blah blah blah. Hail Eris? Hmmm, not sure... wait for rigour first maybe. :)

Knotwork (talk) 09:50, 18 April 2008 (UTC)

Sheesh, I just checked the linked-to glossary of order theory terms, and the word sequence isn't even in their vocabulary! What the heck, how can you even being to approach talking about refining the meaning of the token 'order' in English without making any reference at all to the (in English) seemingly-closely-related token 'sequence' ???????????????? Without even having such a term in your vocabulary/lexicon? Sheesh. What percent of English-speakers might well say "uh, in English we translate that as 'sequence'" ???

Knotwork (talk) 09:59, 18 April 2008 (UTC)

Old discussion
Talk moved from partial order --Markus Krötzsch 17:35, 13 Mar 2004 (UTC)

A general remark for editing order theory topics: please consider using ^ and v instead of the html-special symbols &and; and &or;.

I used to advocate html-math earlier too, but I just changed my browser, finding that it wont display them properly (orders and some arrows work, but all set-theoretic signs are broken). I also looked at other computers, and it did not work there either, so I think it is a general problem these days. The trouble is that on most browsers, you cannot even distinguish the problematic characters from each other. I switched from Mozilla to Firefox (both Linux), but the problem are the (true-type) fonts, not the browser. I checked some IEs too and they did not work correctly either. On one other IE it worked, but the symbols where ugly and way too big for the font size. My old Mozilla was perfect, but it used bitmap fonts which causes other weaknesses. The only things most browsers seem to be able to are &le; and &ge;, maybe also &rarr;, but no more.

The WikiProject Mathematics also recommends to be conservative about these issues, so it is probably a general problem.

--Markus Krötzsch 23:17, 11 Mar 2004 (UTC)


 * In fact $$\vee$$ and $$\wedge$$ do also work nicely (although they may have the wrong size and they require the users of text browsers to read the TeX-Source). --Markus Krötzsch 17:20, 24 Apr 2004 (UTC)

The Alexandrov topology can be defined for any partially ordered set. Here, a set is open iff it is upwards closed. However, there are other topologies of interest for varied types of partially ordered sets, so I doubt that it is "standard". -JB

The most common and easy to read graphical representation of partial orders is in my opinion not DAGs but Hasse diagrams. In this type of diagrams the direction of the order is implied by the relative positioning of the elements. If there is an arc from x to y and y is above x on the paper then x<=y.


 * Would you like to add those two bits of information? Be bold in updating pages :-) --AxelBoldt

Ok, I took the oportunity to add some other things. -JB

Thanks! Could you also explain the notion of "upwards closed subset"? --AxelBoldt

Which relation does "is a subobject of" refer to? -OJarnef

I think it probably refers to relations such as "is a subgroup of", "is a subspace of", "is a subring of" etc.; the term "object" is used in the sense of category theory here. --AxelBoldt

On the page it says "the element u of X is an upper bound for S if s&#8804;u for ALL s in S". Thus an upper bound of S can only exist, if S is TOTALLY ordered, right? Thanks, Thomas

Not right: why should it imply that we can compare s and s' in S, just because we know u is greater than s and s'?

Charles Matthews 14:57, 5 Nov 2003 (UTC)


 * Maybe, for the starters, an example on this: consider the powerset of some set M and take the usual subset ordering. Then M is an upper bound of all elements of the powerset, but these elements are not totally ordered. OK? --Markus Krötzsch 23:17, 11 Mar 2004 (UTC)

In a textbook I am currently reading, "ordered set" is used as short for "totally ordered set" rather than for "partially ordered set". Is this totally idiosyncratic on the part of the author, or is "orderd set" an ambiguous term? Fritzlein 03:56, 28 Jan 2004 (UTC)


 * It is probably a little ambiguous indeed. However, in most contexts of the Wikipedia the intended meaning should be clear. If there are articles that only talk about ordered sets, without using more specific terms, then it might be a good idea to change the wording a bit. --Markus Krötzsch 23:17, 11 Mar 2004 (UTC)

Who actually uses ordered set as shorthand for partially ordered set? -- Walt Pohl 23:32, 13 Mar 2004 (UTC)


 * Both of the standard books given as references to order theory do and probably many others as well. I think this makes sense, since the definition of a mathematical order yields just a partial order. All more special orderings need further qualification. However, mentioning "partial" for clarity at least at the beginning of an article is still a good idea, since some of the more specialized applications may restrict to total orders and use "ordered set" in this sense (which is not a good practice either). I also guess that total orders where considered earlier in history, but todays order theory studies all kinds of partial orders and hence uses "ordered set" in this general sense. --Markus Krötzsch 10:06, 14 Mar 2004 (UTC)


 * The standard meaning of mathematical order is not partial order. If anything, it's total order.  For example, look at ordered field.  It's a total order.  Maybe some books in domain theory and lattice theory use the term ordered set to mean partially ordered set, but in general mathematics usage, it's used to mean total order.  (And I'm not convinced it's all that standard even there.  I just looked at Reynold's Theories of Programming Languages, which talks about domain theory, and he's scrupulous to use partially ordered set.  I also looked at Stanley's Enumerative Combinatorics and an on-line book on universal algebra -- which heavily relies on lattice theory -- and they're always scrupulous to use either "partial ordered set" or "poset".) -- Walt Pohl 18:18, 14 Mar 2004 (UTC)


 * Well, I think the most diplomatic solution is not to consider the term "ordered set" as a strict mathematical concept at all. Usage in different books obviously diverges and is usually not at all confusing if the context is clear. So I think one can continue to use "ordered set" if either an informal intuitive idea of ordering is meant (like in some introduction/motivation sections) or if it has been stated that one really is concerned exclusively with partial or total orders. Formal definitions of course have to be precise (and usually are) but in explanatory texts that follow a definition one does not need to emphasize totality or partiallity of the subjects introduced before (e.g. when saying "For any such order, we find..." or "Some examlpes of ordered sets with this or that property are..."). However, feel free to specify "ordered set" whenever its vagueness is not intentional (as in the introduction to this article). --Markus Krötzsch 09:15, 1 Apr 2004 (UTC)

I would like to give "partially ordered set" its own page, which gives the definition and then links to order theory. The current setup is hard to use for casual users of the definition. -- Walt Pohl 15:22, 16 Mar 2004 (UTC)


 * Yes, I also thought about this already. I think it will be no harm to leave an additional copy of the formal definition within the order theory article, where it fits into the general explanation, since these basic definitions are very unlikely to become inconsistent by independend edits. Just do as you like. --Markus Krötzsch 09:15, 1 Apr 2004 (UTC)


 * OK, done. --Markus Krötzsch 20:29, 27 Apr 2004 (UTC)

Meet operator?
I moved the following text from the article to here for discussion:

Alternatively, the same properties can be described using the notion of meet operator. Meet operator is a function taking two arguments M(a, b) and returning a if a &le; b and b if b &le; a. Using meet operator notation, a partially ordered set is described as follows:


 * M(a, a) = a (reflexivity)
 * if M(a, b) = a and M(b, a) = b then a = b (antisymmetry)
 * if M(a, b) = a and M(b, c) = b then M(a, c) = a (transitivity)

It should be clear that the two notations are equivalent.

I've moved the above here, because I don't think this content about "meet operator" belongs in this article I also have some problems with how it is presented. it could be fixed up and incorporated into meet operator if that article were to be created. I will discuss this further if anyone has any questions or concerns. Paul August &#9742; 16:38, Jan 20, 2005 (UTC)

Article removed from Good articles
This article was formerly listed as a good article, but was removed from the listing because the article lists none of its references or sources --Allen3 talk 20:37, 18 February 2006 (UTC)

GA nomination put on hold

 * GA review (see here for criteria)


 * 1) It is reasonably well written.
 * a (prose): b (MoS):
 * 1) It is factually accurate and verifiable.
 * a (references): b (citations to reliable sources):  c (OR):
 * 1) It is broad in its coverage.
 * a (major aspects): b (focused):
 * 1) It follows the neutral point of view policy.
 * a (fair representation): b (all significant views):
 * 1) It is stable.
 * 2) It contains images, where possible, to illustrate the topic.
 * a (tagged and captioned): b lack of images (does not in itself exclude GA):  c (non-free images have fair use rationales):
 * 1) Overall:
 * a Pass/Fail:
 * Fix 2a,b,c I belive → A z a  Toth 23:17, 7 October 2006 (UTC)
 * Fix 2a,b,c I belive → A z a  Toth 23:17, 7 October 2006 (UTC)


 * Please explain how the article fails 2a (should provide references to all sources) and 2c (reliable sources). -- Jitse Niesen (talk) 04:24, 9 October 2006 (UTC)


 * Can I play Dr. Obvious and point out that the article uses inline cites (2b), too? Lunch 21:41, 9 October 2006 (UTC)
 * Oh, I'm sorry, 2a I can't verify as I can ever know what sources the writer hase used, 2c is ok I see. The thing is 2b that has to be fixed, only the history sections have inlined sources. → A z a  Toth 22:00, 9 October 2006 (UTC)


 * I think this is a good case to examine the effectively of inline cites. So first a rhetorical question what needs to be cited?
 * Much of the article is standard definitions, eg the definition of partial order - citing these individually will simply repeat the same cite over and over again adding unnecessary markup.
 * Alot of the article contains statements which are easily verified by the reader eg, the natural numbers are partially ordered.
 * There are a few no-trivial results, say The finest such topology is the Alexandrov topology, given by taking all upper sets as opens. I guess these are covered by the standard refs.
 * Much of the article is Summary style: There is no need to repeat all specific references for the subtopics in the main "Summary style" article
 * There are two inline cites in the history section, where specific attribute is useful.
 * The whole article is non controversial weakening the need for detailed cites.
 * So it seems the need for many cites is small. The appropriate response might be to follow Wikipedia_talk:Good article candidates where the first statement in the article has an inline cite with a note explaining all statements come from Davey and Priestley. --Salix alba (talk) 22:51, 9 October 2006 (UTC)
 * According to Wikipedia talk:What is a good article?, 2b is a controversal subject, so I hereby decide that 2b is void. and thus this article is a good article. → A z a  Toth 23:00, 9 October 2006 (UTC)

The order it is not the ordered set
In several parts of the article we can find sentences like this one:

'''an element m is minimal if:

a ≤ m implies a = m, for all elements a of the order. '''

That is not precise since the order is a relation on the set, but not the set itself.

The definition could be rewritten, for example:

'''an element m is minimal if:

a ≤ m implies a = m, for all elements a of the ordered set.'''

Discrete linear order
I can't find a definition of "discrete linear order" in Wikipedia.

A discrete linear order is a linear order in which every element except the least has an immediate predecessor and every element except the greatest has an immediate successor. — Preceding unsigned comment added by VictorMak (talk • contribs) 17:04, 11 February 2007 (UTC)

In established usage, a discrete linear order is one where every node has immediate successor or predecessor, unless already maximal/minimal. — Preceding unsigned comment added by 46.7.251.37 (talk) 20:09, 5 August 2021 (UTC)

Some work needed to ensure this is still GA
This article is full of great content, but have a look at the good article criteria and ask if it meets them. It seems to me that it does not meet WP:LEAD, and is a bit weak on citing its sources. The history section could use expansion to meet the broadness criterion, and the cites only support one of its claims. I've improved the formatting of the references, but I think there is more that needs to be done for this article to remain a GA, even in the more relaxed current climate. Geometry guy 21:17, 20 September 2007 (UTC)


 * Unfortunately nothing has changed since last September, so I have decided to reassess the article. The reassessment discussion is transcluded into the section below. Geometry guy 19:20, 29 June 2008 (UTC)

Does there exist a function between orders that is order-preserving but not order-reflecting?
Hi,

in the subtopic Functions between orders the concept of order-embedding is explained as a function that is both order-preserving and order-reflecting. However, this statement is a bit unclear without an example of a function that would be only order-preserving, or only order-reflecting, as the average reader (like I myself) won't be able to find one and be uneasy with the way the definition was formulated. Otherwise this is a very nice article :) Wisapi (talk) 20:24, 17 May 2011 (UTC)


 * I'm not sure it really belongs in the article (hence I'm putting it here) but: let P be the partially ordered set on two elements a and b, with a ≤ b, and let Q be the partially ordered set on two incomparable elements c and d. Then the function that maps both a and b to c is order-preserving but not order-reflecting, while the function that maps a to c and b to d is order-reflecting but not order-preserving.

definition of = in properties
I just re-worked some parts of this article in order to highlight the similarities and differences between preorder, partial order, and total order with respect to the four properties: reflexivity, transitivity, antisymmetry, and conexity.

There was (and is) the following remark in the article:

"Preorders can be turned into partial orders by identifying all elements that are equivalent with respect to this relation."

I tried to elaborate with: "In other words, if one defines a = b if and only if a ≤ b and b ≤ a, then the properties (note that antisymmetry is defined in terms of =) of a partial order are satisfied."

I believe this can be clarified further, possibly by anticipating that the antisymmetric property is itself predicated on another binary relation called "=", and also referred to with "identify" in the remark "by identifying all elements that are equivalent". However, the articles on partial orders, total order, and properties of binary relations themselves do not do this, and they take for granted a unique and "obvious" meaning for "=".

Whereas this remark on preorders becoming partial orders relies on being able to define "=", and so I'm concerned by the possibility of circularity or just plain confusion.

It is likely too formal for this article (and therefore might not help address confusion), but consider something like, e.g. "a set P with two endorelations ≤ and = is a partial order if IsReflexive(P, ≤), IsTransitive(P, ≤), IsConnex(P, ≤), IsAntisymmetric(P, ≤, =). The names, e.g. IsConnex, are abbreviations for a sentence in first-order logic without equality.

It would then be clear exactly what properties need to be satisfied by = (I did not list them above), this might tidy up the whole idea of making a partial order from a preorder.

--Intellec7 (talk) 20:49, 29 January 2021 (UTC)
 * I undid your additions because your (long, rambly, discursive, essay-like, and unsourced) additions (such as one you added into the middle of the definition of partially ordered sets, "A binary relation may have certain properties, and certain groups of properties are given names.") did not (at least, not in my opinion) contribute (or at least, contribute in a positive way) to the readability (and readability is something we should strive for; see WP:TECHNICAL) of the article. —David Eppstein (talk) 21:40, 29 January 2021 (UTC)
 * I suspect you are giving my undue credit for the writing. My edits did not introduce many new sentences, and instead of moved existing language, which I will agree with you is more rambly than I would like. If you have taken a careful look at the edits (https://en.wikipedia.org/w/index.php?title=Order_theory&diff=next&oldid=1003607240), then please give me an example of what is "long, rambly, discursive, and essay-like" that was not previously in the article.


 * I asked you a similar question on your talk page. Please note that this section of this article's talk page is not pertinent to your comments on my edits. --Intellec7 (talk) 21:55, 29 January 2021 (UTC)
 * This section of the article talk page is exactly the right place to discuss changes to the article, including the value of your edits. My personal talk page is not. Also, re "please give me an example": I did give you an example, twice. —David Eppstein (talk) 21:57, 29 January 2021 (UTC)
 * This is not exactly the right section of the talk page, but that is a needless argument to have. If you'd like me to move the discussion from your personal talk page to here, in this section or a new section, just let me know.
 * If I undo your reversion, and remove "A binary relation may have certain properties, and certain groups of properties are given names.", can you tell me what remaining objections you have? --Intellec7 (talk) 22:17, 29 January 2021 (UTC)
 * I have difficulty understanding from your initial comment what you think is wrong with the article that you are trying to improve. For example, confusion about the meaning of "=" only occurs in the context of preorders; by introducing preorders at the same time as partial orders (where the meaning of "=" is totally straightforward) you increased the amount of confusion possible. --JBL (talk) 22:43, 29 January 2021 (UTC)
 * I don't think an article on Order theory should have any particular preference for preorders or partial orders, and what I imagine as an ideal article would not become more confusing by introducing preorders "at the same time" as partial order.
 * "confusion about the meaning of "=" only occurs in the context of preorders"
 * That is true, and so perhaps the comment on how to turn a preorder into a partial order should be omitted, since 1. it introduces confusion, and 2. The intention of my edits is to make it plainly obvious that the only difference between a preorder and a partial order is that the latter satisfies an additional property. If you have a preorder that satisfies this additional property (whether by defining = or using the "totally straightforward" definition), then of course you will have a partial order. --Intellec7 (talk) 23:03, 29 January 2021 (UTC)
 * I have standardized your indentation and placed the quotation in quotation marks; I hope that's all right. I feel we are now awkwardly having the same conversation in two places, but: I disagree strongly about the preference, as I do not believe that order theory (as a discipline) is equally concerned with preorders and partial orders. --JBL (talk) 23:14, 29 January 2021 (UTC)

Highlight similarities and differences among preorder, partial order, and total order
Discussion of this reversion: https://en.wikipedia.org/w/index.php?title=Order_theory&oldid=1003620276 (moved from https://en.wikipedia.org/w/index.php?title=User_talk:David_Eppstein&oldid=1003628096)

I don't think some of these changes ... into the middle of a definition of partially ordered sets

I don't believe that the Order Theory article requires a self-contained definition of any one kind of order (including partial orders or partially ordered sets), since those have their own article. Instead, I believe it should introduce the concepts required for thinking about all kinds of orders. As such, I think it is inappropriate to start from "posets" and then define other orders in terms of adding and removing properties from it. In particular, I do not like the previous version which I summarize as "The familiar notion of order on numbers is a partial order. It satisfies connexity. Indeed it is also a total order."

... changes that muddle the exposition by inserting long unrelated trains of vague thought into the middle of topics that should be more closely related to each other ...

I would appreciate more specificity around your opinion that my edit introduces "long unrelated trains of vague thought". Before my edits the first mention of "preorder" is relegated many sections into the article. My edits have a discussion of total order, partial order, and pre order, which are closely related, located in the same section of the article. --Intellec7 (talk) 21:48, 29 January 2021 (UTC)
 * Maybe I should put it another way. You added roughly 800 bytes of material to the article, but zero new sources. It looked like the material you added was just rambling off the top of your head. Are you trying to bring the article into conformity with what reliable sources say about this topic, or just trying to clarify your own thoughts about it? If the latter, Wikipedia is not really the place for that sort of activity. —David Eppstein (talk) 21:52, 29 January 2021 (UTC)
 * What is the ratio of new bytes to new sources that I should strive for? This edit introduces ~100 new bytes: https://en.wikipedia.org/w/index.php?title=Order_theory&diff=1003599171&oldid=1003596605. 15% of those new bytes are for a new header for "preorder". Some of those bytes are redundant with the existing article, 45% of those new bytes are for (redundant) links to reflexive and transitive relations. 175% of those new bytes (because I removed some things) is for the following sentence:

In other words, if one defines a = bif and only if a ≤ b and b ≤ a, then the properties (which depend on definition of =) of a partial order are satisfied.
 * That sentence elaborates on a point that is unclear (a concern I've already raised on the article's talk page). It cannot be responsible on its own for the article requiring new sources. --Intellec7 (talk) 22:12, 29 January 2021 (UTC)
 * That is very similar to a sentence (already in your previous version) that I (as we were already discussing) reverted: "In other words, if one defines a = b if and only if a ≤ b and b ≤ a, then the properties (note that antisymmetry is defined in terms of =) of a partial order are satisfied." Since you (it seems) have not yet gotten the (many times belabored) point, I don't think that putting parenthetical phrases like "(which depend on definition of =)" into the middle of sentences expressing unrelated thoughts is a helpful writing pattern. —David Eppstein (talk) 00:08, 30 January 2021 (UTC)
 * If you do not like the parenthetical, then you can produce an edit which removes the parenthetical, instead of reverting the edit wholesale. If you think that I've added unsourced information to the article, you can annotate that with a Citation needed tag. The revert is a better option since you have many separate issues with my edit. I am having difficulty making productive use of your comments about those separate issues because your comments have not been sufficiently organized for me to understand them, yet you expect me to already understand them. It is personally unhelpful for you to express your frustration repeatedly, as you have made that abundantly clear. I request you make a stronger effort to assume good faith.
 * I have tried to break out the multiple points that you have issues with as separate sections of this talk page. This point that you have brought up most recently in this thread (and with the goal of minimizing your frustration, let me acknowledge that you have certainly brought it up before) with "the parenthetical phrases like "(which depend on definition of =)"" is pertinent to the section Talk:Order_theory. Can we please try to make progress by discussing that point there? If you want to discuss which of the 800 bytes may or may not need sources, perhaps we can hold that off until I produce another edit for your consideration.
 * I recommend we use this section to discuss the benefits of introducing preorder, partial order, and total order at the same level of hierarchy in the document. Currently, partial order and total order are introduced under '=== Partially ordered sets ===', and preorder is introduced under '== Special types of orders =='. For an example that follows this organizational principle, see this document. --Intellec7 (talk) 19:56, 30 January 2021 (UTC)

Avoid explanations that are relative to partial orders
The article has a certain bias toward partial orders. For example, The visualization of orders with Hasse diagrams has a straightforward generalization: instead of displaying lesser elements below greater ones, the direction of the order can also be depicted by giving directions to the edges of a graph. In this way, each order is seen to be equivalent to a directed acyclic graph, where the nodes are the elements of the poset and there is a directed path from a to b if and only if a ≤ b. Dropping the requirement of being acyclic, one can also obtain all preorders.

Instead of "each order is seen to be equivalent to a directed acyclic graph" + "Dropping the requirement of being acyclic, one can also obtain all preorders", I think it is better to say "each order is seen to be equivalent to a directed graph". And if necessary, the article can elaborate on the correspondence between properties of binary relations and properties of graphs ("antisymmetry" corresponds to "acyclic". "connexity" corresponds to "connected". etc.) --Intellec7 (talk) 22:34, 29 January 2021 (UTC)


 * Ok, this helps clarify things: you think that pre-orders should be the basic object considered in this article. I am not a regular editor of this article and don't plan on continuing in this discussion much further, but let me add my 2c: that seems like a bad idea, not in keeping with the actual nature of the study of order theory.  A supporting data-point: the journal Order gives its editorial board and their interests here; 17 editors mention partial orders or posets, 10 mention lattices, 11 mention some flavor of graph theory, 5 mention some kind of geometry, and 0 mention preorders.  --JBL (talk) 22:54, 29 January 2021 (UTC)


 * I'm afraid you've got it exactly opposite. I do not think that pre-orders or partial orders or any one specific order should be the basic object considered in the article. As the article stands right now, I would say that partial orders serve the role of being the basic object considered. Perhaps there is a good pedagogical reason for this. Perhaps the reason is practical, because as you indicated (thanks for the data point!), partial orders are the most popular kind of order, by some measure. But it is my opinion, and was the idea underlying my recent edits, that the article should NOT have a preference. --Intellec7 (talk) 23:08, 29 January 2021 (UTC)

"<" or ">" as Order relations
Why are "≤" or "≥" used in the article's definition(s)? Regarding two 2 ordered things, one is less than the other, or vice versa. And it's nonsense to ask if something is less or more than itself. — Preceding unsigned comment added by 108.41.98.105 (talk) 19:06, 4 January 2023 (UTC)
 * For total orders, it doesn't make a lot of difference which of these one uses. It is traditional to use ≤ for partial orders and < for (strict) weak orders; I don't know why. But for preorders, the = part of the ≤ relation is not equality, so in that case it is necessary to use ≤, to distinguish the case of two distinct elements that are both ≤ each other from the case of two incomparable elements. —David Eppstein (talk) 19:23, 4 January 2023 (UTC)