Talk:Order theory

message: "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)
thread_title: Initial comments

message: Talk moved from partial order --Markus Krötzsch 17:35, 13 Mar 2004 (UTC)
replies: text: 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)

text: 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≤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)

text: 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)

text: 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)

text: 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)
replies: text:  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)
replies: text:  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)

text: 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)

text: OK, done. --Markus Krötzsch 20:29, 27 Apr 2004 (UTC)
thread_title:  Old discussion 

message: 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)
thread_title: Article removed from [[Wikipedia:Good articles]]

message: 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)
thread_title:  Discrete linear order 

message: 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)
replies: text: 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)
thread_title:  Some work needed to ensure this is still GA 

message: 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)
replies: text: 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)
replies: text:  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)
thread_title:  Avoid explanations that are relative to partial orders 

message: 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)
replies: text: 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)
thread_title:  "<" or ">" as Order relations