Talk:Order theory

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History needs a bit more work, but that's it. Tompw 15:03, 5 October 2006 (UTC)

Contents

[edit] 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 ∨.

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 ≤ and ≥, maybe also →, 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≤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)

[edit] 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 ≤ b and b if b ≤ 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 ☎ 16:38, Jan 20, 2005 (UTC)

[edit] Article removed from Wikipedia: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)

[edit] GA nomination put on hold

GA review (see here for criteria)
  1. It is well written.
    a (prose): b (structure): c (MoS): d (jargon):
  2. It is factually accurate and verifiable.
    a (references): b (inline citations): c (reliable): d (OR):
  3. It is broad in its coverage.
    a (major aspects): b (focused):
  4. It follows the neutral point of view policy.
    a (fair representation): b (all significant views):
  5. It is stable.
  6. 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):
  7. Overall:
    a Pass/Fail:


On Hold. Fix 2a,b,c I belive →AzaToth 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. →AzaToth 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 Wikipedia: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#Can a compromise be found? 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?#Citation method, 2b is a controversal subject, so I hereby decide that 2b is void. and thus this article is a good article. →AzaToth 23:00, 9 October 2006 (UTC)

[edit] 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.

[edit] 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.

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