Talk:Interval (mathematics)

Introduction

Doesn't the definition of Interval make as much sense for a poset as a totally ordered set? There is a link to this page from poset that would suggest as much. I'll make the change and expect someone to let me know if I am way off base. ;-> -- Jeff 18:20 Jan 22, 2003 (UTC)

Ah, just changing the definition to use posets allows incomparable elements to be an "Interval" which just wouldn't be right. So I guess the reference from poset is really pointing to the interval notation which can be extended to posets (and just might produce empty sets a lot). Is this used enough to be worth mentioning? -- Jeff

I added [a,b] for a partially ordered set - Patrick 23:00 Jan 22, 2003 (UTC)

In the section about Interval Arithmetic... Division by an interval containing zero is indeed possible if some extensions are made. This makes it possible to get answers such as $[a,\infty ]$.

Divsion with intervals may result in two intervals. Ex. say [1,1]/[-1,1] results in two intervals. [-inf,-1] and [1,inf]

Varying vs. constant interval

Let's say P₁ = Q₁ = 100 and P₅ = Q₅ = 500, but P₂ = 197, P₃ = 301 and P₄ = 404, while Q₂ = 200, Q₃ = 300 and Q₄ = 400——i.e., the "P" and "Q" endpoints are the same, while the "Q"s are equally spaced between each other, and the "P"s aren't. For example (TN = "term number" and UT = upper TN):

$P_{tn}=F(x)G(y_{tn})H(z)\,\!$

while

$Q_{tn}=Q_{1}+{\frac {TN-1}{UT-1}}\Delta Q=P_{1}+{\frac {TN-1}{UT-1}}\Delta P\,\!$

Would the proper definition/identification be "the Q's provide auxiliary points between P₁ and P₅ at a constant interval"? Or does this concept have an established name? This article doesn't appear to address this variation.  ~Kaimbridge~20:01, 16 February 2006 (UTC)

fixed improper examples of open and closed intervals

The reason I'm making a change: There are many things wrong with the text I am replacing, but mostly it's because by definition an interval is a set which contains two endpoints so a single valued set can not be an interval. The previous definitions for open and closed intervals also looks more like a discussion of numbers dating back to the turn of the previous millennia where scholars where discussing odd and even numbers more as a philosophical problem. Open and closed intervals have nothing to do with single valued sets nor whether [integers] are open or closed.

Reference: "Calculus With Analytic Geometry" by Earl W. Swokowski, Prindle, Weber & Schmidt, 1979, ISBN: 0-87150-268-2. Pages 5 and 6.

I've removed the following: Intervals of type (1), (5), (7), (9) and (11) are called open intervals (because they are open sets) and intervals (2), (6), (8), (9), (10) and (11) closed intervals (because they are closed sets). Intervals (3) and (4) are sometimes called half-closed (or, not surprisingly, half-open) intervals. Notice that intervals (9) and (11) are both open and closed, which is not the same thing as being half-open and half-closed.

Intervals (1), (2), (3), (4), (10) and (11) are called bounded intervals and intervals (5), (6), (7), (8) and (9) unbounded intervals. Interval (10) is also known as a singleton.

The length of the bounded intervals (1), (2), (3), (4) is b-a in each case. The total length of a sequence of intervals is the sum of the lengths of the intervals. No allowance is made for the intersection of the intervals. For instance, the total length of the sequence {(1,2),(1.5,2.5)} is 1+1=2, despite the fact that the union of the sequence is an interval of length 1.5.

and added the following: Intervals using the round brackets ( or ) as in the general interval (a,b) or specific examples (-1,3) and (2,4) are called open intervals and the endpoints are not included in the set. Intervals using the square brackets [ or ] as in the general interval [a,b] or specific examples [-1,3] and [2,4] are called closed intervals and the endpoints are included in the set. Intervals using both square and round brackets [ and ) or ( and ] as in the general intervals (a,b] and [a,b) or specific examples [-1,3) and (2,4] are called half-closed intervals or half-open intervals.

Rockn-Roll

Question for Functional spaces

Let be I a set in functional space of all the functions f(x) defined on the interval $[0,1]$ my question is if we can get some $I_{1},I_{2},I_{3},......$ subsets of I, or if the case is more complicate dealing with numbers than with functions.--85.85.100.144 09:11, 19 February 2007 (UTC)

Types

In the section "Higher mathematics," intervals that are closed at infinity should be mentioned as well. (e.g. $\left[\infty ,-\infty \right]=\mathbb {R} ^{+}$ the extended reals.)— Preceding unsigned comment added by He Who Is (talk • contribs) 23:02, 9 June 2006

This seems done now. OTOH in sect. "infinite endpoints" there is the confusing paragraph "The notations [−∞, b] , [−∞, b) , [a, +∞] , and (a, +∞] are ambiguous (...etc...)", while in a subsequent section ("extended real line") the notation is defined. Wouldn't it be better to delete the first paragraph (or only leave "For authors who define intervals as subsets of the real numbers, the notations .... are either meaningless or equivalent to ...") -- either there or maybe (better?) after the definition? — MFH:Talk 21:25, 8 April 2012 (UTC)

It will be better to realize the desired output before changing the existing text. There are actually three possible approaches to [a, +∞] and similar notations:

1. "[−∞" and "+∞]" are ambiguous without specifying the domain, and are equivalent to "(−∞" and "+∞)" respectively if the domain defined to be the real line;
2. [a, b] is a subset of extended real number line if a = −∞ or b = +∞;
3. "[−∞" and "+∞]" imply extended real number line if an infinity is noted explicitly, but defaults to the real line if it is not.

All three approaches have advantages and disadvantages. (1) is better when we need a parameter-dependent family of closed intervals on ℝ (but where parameter(s) can go to infinity), and is confusing in all other situations. (2) is fairy unambiguous where "−∞" or "+∞" are noted explicitly, but forces us to specify either inclusion or exclusion of infinities each time when we define "an arbitrary interval [a, b]". (3) appears to be the most convenient, but is a quite complex convention. Is it known which approach prevails in textbooks? Incnis Mrsi (talk) 09:51, 9 April 2012 (UTC)

Interval Arithmetic in Fortran and C++

Should it be noted that the Sun Studio compilers implement Interval Arithmetic? --rchrd 03:13, 11 July 2006 (UTC)

Done.--Patrick 09:39, 22 May 2007 (UTC)

Use of German Wikipedia Entry as a "See Also"

It should be considered bad form to ask the user to reference another language edition of Wikipedia for more information, can anyone correct this? Anpheus (talk) 19:32, 20 February 2008 (UTC)

Perhaps link to a google or babelfish translation.Tailsfan2 (talk) 16:19, 12 May 2008 (UTC)

Rewrite required

I see several serious issues with the article, and I think it requires a comprehensive revision. Before going ahead and implementing this, I thought that I'd start a discussion. Following are problems with this article:

• Lede: The definition given excludes unbounded intervals such as $[0,\infty )$.
• Also: interval is not a concept from Algebra. It has more to do with analysis, geometry and topology than with algebra.
• The section labeled informal definition basically talks about notation, not definition.
• The section labeled Formal definition speaks about intervals in ordered sets. When mathematicians speak of intervals, they usually mean intervals of real numbers. The mathematical definition of an interval is just a convex subset of the real numbers.

Often, maybe. Usually, can't say. Always - absolutely NOT. SteveWoolf (talk) 05:38, 14 November 2008 (UTC)

• There is another section on intervals in partially ordered set, which should be merged with the Formal definition section.
• It is unclear to me what the operation $x\cdot y$ means in the section on relational operations. Therefore, the whole section is unclear.

Oded (talk) 23:24, 15 April 2008 (UTC)

Seems like a good idea to me. If I have time, I'll post with some more details. But in the mean time, my suggestion is to be BOLD! Cheers, silly rabbit (talk) 23:29, 15 April 2008 (UTC)

Rewrite accomplished. Oded (talk) 01:45, 21 April 2008 (UTC)

Why not define for ANY poset?

The concept of interval is both meaningful and useful for partially ordered sets in general. This should top the article. Then give Real numbers as a special case. As for the case of non-comparable elements a and b, the interval [a,b] will be ø, the empty set. SLWoolf (talk) 16:25, 30 October 2008 (UTC)

I agree to the extent that the general definition should be mentioned in the lede, not exclusively in a tiny section towards the end of the article. On the other hand, intervals of real numbers is what the vast majority of our users will expect here. And everything interesting that can be said about intervals in order theory really belongs into other articles anyway. Therefore in my opinion a single sentence at the end of the lede, pointing away to the appropriate article (partially ordered set) is enough. Not sure if I have the time right now; if I do, I will try to fix the lede and the little paragraph further down. --Hans Adler (talk) 19:09, 30 October 2008 (UTC)

OK, it's slightly more work than I thought because we need to change the definition (case by case for open, closed and half-open intervals). So I can't do it now. --Hans Adler (talk) 19:24, 30 October 2008 (UTC)

You don't have to change any definitions - but they could be much clearer. See the definition of Interval in Partially ordered set. In any case, your article is primarily about REAL intervals and should be renamed that. Not to do so is both incorrect and problematic for those of us who work with partial orders - we need to link to a general Interval definition. S0 - as you are busy, I have tidied things up for now - but technically this article should be renamed. An article specifically about gorillas, for example, should not be named "Great Apes". Hope you agree! Regards SteveWoolf (talk) 05:29, 14 November 2008 (UTC)

The title of the article doesn't have to be perfect, as long as it's easy for the reader to find the article they actually want. In this case, the note at the beginning that points to the article on posets should be enough to clarify the subject of this article. — Carl (CBM · talk) 14:54, 14 November 2008 (UTC)

As I said once, an article specifically about gorillas, for example, should not be named "Great Apes". It isn't correct to call an article Interval and then only talk about a subset of the subject. Of course, I realise renaming it is alot of work, since it has so many links to it. SteveWoolf (talk) 16:30, 14 November 2008 (UTC)

On the other hand, as a mathematician, almost every time I say "interval" I mean "real interval"; this is probably true for everybody except order theorists. And readers who come to the article are most likely to be looking for the definition from their algebra or calculus class. So it's not quite the same situation here as with great apes, and I think that "isn't correct" is too strong a phrase for what's going on. As a somewhat parallel example, the article on concrete is about concrete bound with cement, even though asphalt concrete is also a type of concrete. — Carl (CBM · talk) 16:49, 14 November 2008 (UTC)

Okay, Maybe you're right. Be interesting if we actually knew what % of hits on the article were Real Intervals; but as you say, probably most. I'm happy with the article as it now is, so lets concrete it in and keep the apes out. (Just kidding) SteveWoolf (talk) 07:24, 17 November 2008 (UTC)

Recent Vandalism

To whoever anonymously wrote how he hates Calculus: If you vandalise again, you will be reported. Why not try to make a postive contribution to WP on a topic that interests you? Even sex, drugs and rock'n'roll are all valid topics. I also don't care much for calculus, but I expect you like using things whose production requires advanced mathematics, such as cars and planes and DVD's. SteveWoolf (talk) 12:08, 22 November 2008 (UTC)

Error / Unclear Sentence in Notation for intervals section

In section Notation for intervals subsection Excluding the endpoints formatted math text:

${\begin{aligned}(a,b)&=&]a,b[&=\{x\in \mathbb {R} \,|\,a<x<b\},\\{}[a,b)&=&[a,b[&=\{x\in \mathbb {R} \,|\,a\leq x<b\},\\(a,b]&=&]a,b]&=\{x\in \mathbb {R} \,|\,a<x\leq b\},\\{}[a,b]&&&=\{x\in \mathbb {R} \,|\,a\leq x\leq b\}.\end{aligned}}$

Does not agree with the last sentence of the subsection Excluding the endpoints

Some authors use $]a,b[$ to denote the complement of the interval (a, b); namely, the set of all real numbers that are either less than or equal to a, or greater than or equal to b.

I don't know if the last sentence is to illustrate that not all authors follow the set builder notation prescribed by International standard ISO 31-11, or if it is an error an should be deleted to avoid confusion, since the meaning of $]a,b[$ is already covered.— Preceding unsigned comment added by ‎8.30.81.10 (talk) 19:52, 28 December 2012

I don’t see what set-builder notation has to do with the issue, but it’s clearly the former (i.e., a warning that some authors use the perverse bracket notation with a different meaning).—Emil J. 20:25, 28 December 2012 (UTC)

I agree that this sentence is very strange and confusing. I never saw anybody use this notation like this ; also there is no reference for this sentence, so why not removing it ? Also I do not understand why Emil J called this bracket notation "perverse" ; maybe he wanted to say "reverse" ?? --89.95.99.135 (talk) 11:05, 24 November 2018 (UTC)

Usage of open-ended integer intervals

According to the article

Alternate-bracket notations like [a .. b) or [a .. b[ are rarely used for integer intervals.

This statement is false (note that it has no citation), as in fields like computer science it is extremely common to note indices in this way. One such example is in the article one Zero-based_numbering which uses the integer range [0, n).

50.1.107.248 (talk) 01:01, 26 August 2014 (UTC)

Additionally, Python's `range` is `[)` and Rust has `..` which is `[)` and `..=` which is `[]`. Solomon Ucko (talk) 12:20, 11 February 2020 (UTC)

Examples for infinite endpoints backwards?

Both references to "[−∞, b] , [−∞, b) , [a, +∞] , and (a, +∞]" look like the change in notation doesn't match what the next is explaining; shouldn't it switch between open and closed for +/-∞ rather than a and b, since it's talking about the behavior towards +/-∞ ? Dirk Gently (talk) 13:10, 6 September 2014 (UTC)

Consider what the article is saying more closely. These are all meaningless (and not to be used) in the context of the real numbers. However, in the extended real number line, the two infinities are actual elements (and not statements about limits), so including or excluding them makes sense. Bill Cherowitzo (talk) 13:50, 6 September 2014 (UTC)

A closely related but different question: the article makes the statement

The notations [−∞, b] , [−∞, b) , [a, +∞] , and (a, +∞] are ambiguous. For authors who define intervals as subsets of the real numbers, those notations are either meaningless, or equivalent to the open variants. In the latter case, the interval comprising all real numbers is both open and closed, (−∞, +∞) = [−∞, +∞] = [−∞, +∞) = (−∞, +∞] .

Why would we make the statement that "equivalent to the open variants"? While some sloppiness in notation might occur, it surely makes no sense to document the closed variants on the real line as other than meaningless? And the claim of ambiguity also simply makes no sense? —Quondum 18:31, 6 September 2014 (UTC)

I agree, there is something a little fishy here. I find nothing ambiguous; if you are not talking about the extended real line, using the closed endpoint in these situations is just wrong. Also, in the real line case, those variants of (-∞, +∞) seem to be confounding the difference between closed interval and closed set and I suspect that no reliable source could be found for them. Bill Cherowitzo (talk) 13:40, 7 September 2014 (UTC)

I've updated the claims. I don't quite follow what you're saying about the real number case, though. —Quondum 20:13, 7 September 2014 (UTC)

A suggestion for consistency, helpful for non mathematicians

Compared to the substantive discussions above, this change, if it would be valid, would be just for consistency; but wouldn't it be better, so as to be more consistent with all of the other nearby examples in the section Classification of intervals, in the examples for left-bounded and right-unbounded, to reverse the positions of x and a, that is, to say a < x and a ≤ x ? Wikifan2744 (talk) 00:34, 23 September 2014 (UTC)

Merger proposal from Values interval

I propose to merge the newly created stub values interval into this article (Interval (mathematics)). The content of that article appears to be talking about ordinary number intervals of the sort that this article already covers quite well, only couched in some of the language of signal processing. I'd love to get some input from someone knowledgeable about signal processing to clarify whether the phrase "values interval" is actually in common use in that (or any other) discipline, since it's not one I've ever heard before, and whether there's anything notably different about the way an interval of values is thought of, used, or described in that particular field to merit a separate article. In the absence of some clarification, I consider values interval a duplicate and would like to redirect it here.-Bryanrutherford0 (talk) 02:18, 24 September 2016 (UTC)

I have to admit that I forgot what I wrote there (and it seems no longer accessible), but I remember that this was meant to help people from various background to understand better some aspect of the dynamic range, which is also in the end some relative of this very article for signal analysis folks. Probably some content from there might be useful in this article, to help clarify the relationship between dynamic range and values interval (range). Riccardo.metere (talk) 06:15, 24 September 2016 (UTC)

Probably, it should be somewhat referenced to Range (statistics) and Range (mathematics) as the concepts are sometimes overlapping but the term used around are always the same... Riccardo.metere (talk) 08:23, 24 September 2016 (UTC)

It looks like it's been redirected by someone else.-Bryanrutherford0 (talk) 12:54, 24 September 2016 (UTC)

Dyadic interval clarification

Dyadic interval is defined in the text as [ k/2^n, (k+1)/2^n ] for some integers k, n. Maybe my brain isn't working today but as far as I can see such an interval is always of length 1 which makes little sense and contradicts what's stated in the paragraphs that follow after the def. Maybe someone could be kind enough to clarify. 84.219.174.71 (talk) 11:49, 13 January 2017 (UTC)

Could use a little section on international colloquial notation

5...7

5–7

5~7 — Preceding unsigned comment added by 2001:8A0:5E59:9F00:F092:4F6:C203:2042 (talk) 22:25, 7 December 2020 (UTC)

ultra irrational numbers

I am a student of class ten and I was doing math yesterday morning and a sudden I was faced of with a number that I named ultra irrational numbers, if I say you to give the root of 2 or 3 or 6 you wont be able, but can give an approximation but I will give you to write the root of a number but you wont be able to write a single line,; root -2, I know that no one can write a single line after this question, I named it ultra irrational number because it looks like irrational numbers but these are unreal numbers, so I could be able to write 5 types of unreal numbers; infinity, -infinity, √-2, infinitesimal, -infinitesimal MAKRI234 (talk) 11:03, 16 September 2022 (UTC)

I suggest you read about Complex numbers. Dhrm77 (talk) 13:15, 16 September 2022 (UTC)

Mathisfun

Mathisfun is cool but this article needs more reliable sources .e.g books. AXONOV (talk) ⚑ 07:50, 29 March 2023 (UTC)

Mass-revert

@D.Lazard: That was too careless; among what you have removed there were some new stuffs, such as results on linera continua, order-convexity, etc. Besides, please try to be specific about what you are going to criticize, "pedantic" is a little vague and rough to me. 慈居 (talk) 09:23, 29 August 2023 (UTC)

By "pedantic", I mean restating things that are alredy in the article in a way that is more difficult to understand to non-specialists.

"New stuffs" are not allowed on Wikipedia per the policy WP:OR, unless they are supported by reliable sources, which is not the case here.

Moreover, this article is about intervals of real numbers. So, general results on order-convexity do not belong to this article.

By the way, your general definition of intervals in preordered sets seem to be original research, as not occurring in any textbook that I know. Moreover, you give a definition such that the intersection with $\mathbb {Q}$ of a (real) interval is not necessary an interval of $\mathbb {Q} .$ This goes against the common usage. So, I'll remove the part of your edit that I moved to section § Generalizations. You can restore it if you are able to provide a reliable source (that is a textbook, since this is an elementary article). D.Lazard (talk) 10:43, 29 August 2023 (UTC)

"New stuffs" are not allowed on Wikipedia per the policy WP:OR, unless they are supported by reliable sources, which is not the case here.

I have aleady provided some necessary references for the "new stuffs". I referred to those stuffs as "new" because you claimed that they were duplicates.

this article is about intervals of real numbers

No, this article was more than just about real intervals even before my edits. The hatnote clearly stated that "This article is about intervals of real numbers and other totally ordered sets". There are also a a section for integer intervals, which are intervals in the totally ordered set of integers, and a section for intervals in the extended real line, which is a totally ordered set and a complete lattice.

By the way, your general definition of intervals in preordered sets seem to be original research, as not occurring in any textbook that I know.

Check .

Moreover, you give a definition such that the intersection with $\mathbb {Q}$ of a (real) interval is not necessary an interval of $\mathbb {Q} .$ This goes against the common usage.

Go argue here. 慈居 (talk) 11:37, 29 August 2023 (UTC) 慈居 (talk) 11:37, 29 August 2023 (UTC)

References

New edits by 慈居

User 慈居 has recently completely changed the article. As these changes are highly controversial, I'll restore the previous stable version. As it is the rule on Wikipedia (see WP:BRD), none of these changes must be done again if a consensus is not reached on this talk page. Someof the controversial points of these edits are the following

• Replacement of the standard definition of intervals by a definition based on the endpoints. This leads to split the definition in multiple cases, in contrast of the usual definition, for which these cases are properties of specific intervals.
• Change of the scope of the article by introducing before the description of the properties of real intervals a wide generalization to preordered sets. This article must focus on real intervals, since, for most people, interval means real interval.
• The generalization of intervals on preordered sets that is introduced contradicts the usual definition for totally ordered sets and for partially ordered sets given in Partially ordered set#Intervals. In particular, the set of rational numbers whose square is less than 2 is explicitly considered as not being an interval of rational numbers.
• The systematic use of set builder notation, where prose would be more readable (the given general definition of intervals consists of 9 lines of set builder notation without prose between them).

The article was not a good article, but, IMO, with 慈居's changes, it is much worse. D.Lazard (talk) 10:39, 4 September 2023 (UTC)

@D.Lazard:

Replacement of the standard definition of intervals by a definition based on the endpoints.

The original definition is acutally the definition of (order-theoretical) convex sets, which is weaker than being an interval, even for totally ordered sets. I have reserved the original definition for real intervals since in the real line, it is equivalent to the standard definition (based on endpoints, in your words). I don't see what bothers you.

[F]or most people, interval means real interval.

I agree.

This article must focus on real intervals, since, for most people, interval means real interval.

I don't agree. Intervals in ordered sets are a notable concept in some branches of mathematics (e.g. closed intervals occur frequently in lattice theory). Some people (including me) need those informations. And again, I didn't remove any contents about real intervals, so I don't see what bothers you. If my edits confuses someone, clarify by saying something like "in many contexts, intervals mean real intervals by default", not just blow up everything.

The generalization of intervals on preordered sets that is introduced contradicts the usual definition for totally ordered sets and for partially ordered sets given in Partially ordered set#Intervals.

At least two references say the opposite. (Those were in my edits and you should have checked them before commenting.) A one-sentence quote will suffice for this discussion. "Clearly every interval in $X$ is convex but not conversely, and we will, as usual, denote intervals by $(a,b)$, $(a,b]$, $[a,b)$, or $[a,b]$." (From )

The systematic use of set builder notation, where prose would be more readable (the given general definition of intervals consists of 9 lines of set builder notation without prose between them).

Again, not a reason for a mass-revert. Actually, I was going to add some symbol-free explanations, but you reverted my edits too early.

WP:BRD is not an excuse for careless removal of referenced contents. You haven't explained yet why you have removed the equivalence between intervals, (order-theoretical) convexity, (path/arc) connectedness, the results about linear continua, and the introduction of convex sets and convex components. Wikipedia helps not only high school students but also thousands of math researchers. This is why elementary and advanced topics can and must coexist. 慈居 (talk) 18:08, 4 September 2023 (UTC)

In any case, your edits are a major change of the article, and as such require a consensus on the talk page. In any case, the stuff on ordered sets does not belong to this article. It belongs either to partially ordered set#Intervals or to an article that you are free to submit and could be called Intervals on preordered sets or Order convexity. D.Lazard (talk) 18:40, 4 September 2023 (UTC)

@D.Lazard: Originally, the article covered intervals in totally ordered sets. It's your recent edits which restrict it to real intervals. Why not for posets and preordered sets? 慈居 (talk) 18:45, 4 September 2023 (UTC)

"Originally, the article covered intervals in totally ordered sets": This has not been changed, since I have not removed the sententence in the lead, and I have added a definition in § Generalizations. I have removed the mention in the hatnote, since totally ordered sets are are specific posets.

"Why not for posets and preordered sets?": because Wikipedia is written for a large audience, not for professional mathematicians. Most people who need information on intervals need it in the context of calculus, and many ignore posets and preordered sets. Also, reals intervals are historically known since centuries, and the generalization to posets is not older than the definition of posets, that is during the 20th century. So, the generalization to posets is normally placed in Partially ordered set, with a mention in this article in section § Generalizations.

About the generalization to preordered sets: it must be well established for being acceptable in Wikipedia. So, you have to provide an evidence of this well-establishment. A research article giving a definition of intervals that is specific to this research article cannot be used to provide such an evidence. D.Lazard (talk) 20:32, 4 September 2023 (UTC)

@D.Lazard:

Wikipedia is written for a large audience, not for professional mathematicians

Adding stuff doesn't exclude the audience. Removal does.

Most people who need information on intervals need it in the context of calculus, and many ignore posets and preordered sets. Also, reals intervals are historically known since centuries, and the generalization to posets is not older than the definition of posets, that is during the 20th century.

It suffices to give the real intervals some priority. (I have already done so, by introducing real intervals first in each section.) Also, intervals in totally ordered sets are covered in the article, despite also relatively new.

[...] with a mention in this article in section § Generalizations.

Not until your recent edits.

About the generalization to preordered sets: it must be well established for being acceptable in Wikipedia. So, you have to provide an evidence of this well-establishment. A research article giving a definition of intervals that is specific to this research article cannot be used to provide such an evidence.

No. By the policy guideline, a reliable source is enough and I have provided several. 慈居 (talk) 20:51, 4 September 2023 (UTC)

The fact that there is a definition means that it is used in that book. 慈居 (talk) 20:58, 4 September 2023 (UTC)

I want to add that, actually an enourmous portion of mathematics is done in 20th century. It makes no sense to split them from the older stuffs. 慈居 (talk) 21:15, 4 September 2023 (UTC)

@D.Lazard: Do you have any other comments? 慈居 (talk) 05:16, 14 September 2023 (UTC)

My concerns given at the beginning of the section remain completely valid, and I continue to disagree with your edits. As no third-party opinion has been given, I'll ask help at WT:WikiProject Mathematics for resolving this content dispute.

I see that you restored your edits. I'll revert them to the previous stable version until a consensus is reached (in case of no consensus, Wikipedia rule is that is this is the version that prevails). D.Lazard (talk) 09:23, 14 September 2023 (UTC)

• I think that the previous version should stay, and the reasons why derive from WP:NPOV and WP:WEIGHT. Changing an article to discuss the most general possible presentation of a concept, rather than the best-known one, is a common instinct, but it is incorrect and inconsistent with content policy. According to WP:WEIGHT, we need to present the various perspectives on any subject in proportion to their coverage in the literature, regardless of which is more general or all-encompassing. Therefore, because of the overwhelming coverage of real intervals rather than the most abstract order theoretic definition, it is not enough that the new edits didn't remove any contents about real intervals, and it would not be enough to [say] something like "in many contexts, intervals mean real intervals by default". Because we need to use due weight, presenting all the highly technical stuff about intervals in posets and preorders, etc., alongside real intervals, is not okay, even if none of the material about real intervals is deleted. This also refutes the arguments that Adding stuff doesn't exclude the audience. Removal does and a reliable source is enough and I have provided several.

If there is going to be content about the general formulation of intervals in the context of preorders and so on, it should be contained in its own section here or split off into a separate article. This would also address the concern that elementary and advanced concepts can and must coexist: the abstract formulations would still be on Wikipedia, just not in a way that violates core content policy. ByVarying | talk 15:12, 14 September 2023 (UTC)

@ByVarying: These edits are a tentative for clarifying the focus of the article on real intervals. They include the removal of section § Topological algebra, see next thread). They were reverted for restoring the stable version. Third party opinions would be welcome, about whether this improves the stable version. D.Lazard (talk) 15:56, 14 September 2023 (UTC)

I think these edits are fine. ByVarying | talk 21:57, 15 September 2023 (UTC)

The basic scope/content of an article called "Interval" should be real intervals. There is enough to say about the topic of real intervals to fill an arbitrarily long Wikipedia article, without needing to expand the scope. Other generalizations should be relegated to a (relatively short) section near the end on "Generalizations" or the like, more of a mention than a full description.

User:慈居: if you think the more general concept is also important to cover, you should make it into an entirely separate article, titled something like Interval (order theory) (or pick your favorite title that covers the scope you care about), and link it from the Generalizations section here using the {{main}} template. But including a lot of material about a generalized topic in the early part of this article is out of scope. It makes the article less readable for the intended wide audience because it is a distracting irrelevance and is unnecessarily technical. –jacobolus (t) 16:35, 14 September 2023 (UTC)

Is this in reply to me? That's exactly what I am saying. ByVarying | talk 17:02, 14 September 2023 (UTC)

Yes, I am agreeing with you and expanding on the same point. Though I think it's less a matter of due vs. undue weight, and more a matter of (a) keeping the scope coherent and putting largely separate topics into their own space, and (b) not making articles aimed at a general audience more technical than necessary. –jacobolus (t) 17:08, 14 September 2023 (UTC)

I appreciate everyone's comments and I will not insist on restoring my version until finishing the discussion. But I do think that it's a bad manner to respond after ten+ days of silence (while still active in Wikipedia) according to which I had decided to restore my edits.

I don't think my orginazation of contents causes undue weight. Although used relatively less, the concept of intervals in general has its own significant noteworthiness. I'd agree it has undue weight if it is an incorrect or deprecated theory, but it's just not. A similar case is that the circles are Euclidean circles by default, but there are also noteworthy notions of hyperbolic circles, and they have a unified definition in Riemannian geometry.

The distinction I wrote between intervals and order convexity is what people have been asking a lot in online math communities ([1], [2], [3]). I mean, some people need what I wrote! They are possible readers of Wikipedia and don't deserve to be neglected just because they're not many. (Actually a lot of questions in the commmunity quote Wikipedia, which indicates that they might not have asked if that's explained here.)

The original version of the article covered integer intervals, also intervals "with infinite endpoints". It will be natural and useful to include a unified definition which explains e.g. why the integer interval from one to ten does not include two and a half, and which gives insights on how to treat intervals in the extended real line not just as special notations (as the current article does). So I disagree that what I wrote is of "distracting irrelevance". It's more of a completion.

@Jacobolus: I do agree with your readability concerns. However, I still insist that a summary in a tiny little section is not enough, based on the reason I have provided above. Personally I like the structure of limit of a sequence, in which various layers of generalizations of the concept is covered, each having fair treatment. How about moving what I wrote to a section Intervals in ordered sets? I have a lot to write and a subsection of Generalizations is really not enough.

@D.Lazard: Again, you haven't explained some of your removals. You have also removed my edits about real intervals (and totally ordered sets, which is a topic the original article covers and claims to cover). See Properties section of my version. 慈居 (talk) 04:49, 15 September 2023 (UTC)

it's a bad manner to respond after ten+ days of silence – this page was not previously on my watchlist, so I only noticed this conversation when D.Lazard brought it to the wikiproject mathematics talk page. Wikipedia is an indefinitely running volunteer project with many contributors who go months or even years between contributions; there is no short-term deadline involved here, nor is a few days enough time to establish community consensus. It is not "bad manners" to respond at any particular time. (If someone wants to revisit this topic in, say, 3 years, it will not be bad manners for them to do so.)

A similar case is that the circles are Euclidean circles by default, but there are also noteworthy notions of hyperbolic circles, and they have a unified definition in Riemannian geometry. – I would eventually like a paragraph or two about this to Circle § Generalizations down near the bottom of the page. I tried to somewhat improve the article about spherical circles though it still has a long way to go. A general article about circles or cycles in hyperbolic geometry would be good to add (we currently have horocycle and hypercycle, but they are a bit disjointed). Adding another separate article about circles on Riemannian manifolds sounds like a great idea. But by no means should all of these topics be covered in full detail on the article circle. It's way out of scope.

concept of intervals in general has its own significant noteworthiness [...] tiny little section is not enough [...] – this is why you should create an entirely new article with a clear scope. –jacobolus (t) 05:09, 15 September 2023 (UTC)

I apologize if I phrased it wronng, but I referred to the particular user User:D.Lazard when I say "bad manner". I got response from this user only after I restore my edits, not having seen any replies from the user since 4 of September while the user is active elsewhere in English Wikipedia. I think I have a reason to feel annoyed. 慈居 (talk) 05:59, 15 September 2023 (UTC)

It was clearly stated that comments by other editors were needed before further editing. Otherwise, I had nothing to add to what I already wrote, and it is useless to discuss specific edits when there is such a fundamental disagreement on the scope and the general organization of the article. D.Lazard (talk) 06:19, 15 September 2023 (UTC)

Otherwise, I had nothing to add to what I already wrote, [...] Maybe you could have said that earlier? When I ask you whether you have any comments, your only response before my restoring was to archive my comment in your talk page, and I thought we're done. 慈居 (talk) 06:49, 15 September 2023 (UTC)

Your only response before my restoring was to archive my comment: I am not responsible of the automatic archives. This has been done by a bot. Moreover, this archived comment on my talk page was an invite to discuss on the article talk page, and I answered here in section #Mass revert, despite its unfair heading. D.Lazard (talk) 07:43, 15 September 2023 (UTC)

I am not responsible of the automatic archives. Okay. That was my misunderstanding. I thought the edit was done manually because my comment was archived while the previous talks are still there.

[…] despite its unfair heading. I too think that the heading was emotional, but that was my reaction to the whole removal of my edits including well referenced parts without discussion. People don't do that normally, even if they disagree with the edits. 慈居 (talk) 08:01, 15 September 2023 (UTC)

And I still think that it's an overkill to remove all my edits for disagreement on the article structure. 慈居 (talk) 08:13, 15 September 2023 (UTC)

@Jacobolus: What do you think about the structures of Limit of a sequence, Continuous function or Series (mathematics)? There still can be separate articles for specific aspects of the concept (e.g. continuous map between metric spaces, series of functions, etc.), even if the parent article covers all of them in necessary details. And hopefully the separate articles are created by editors other than those of the main articles, in their own phrasings and styles. 慈居 (talk) 07:32, 15 September 2023 (UTC)

I have created this page to illustrate what I have in my mind. 慈居 (talk) 12:41, 15 September 2023 (UTC)

This kind of organization is definitely better than trying to put the stuff about intervals in posets etc. up near the top. @D.Lazard what do you think? –jacobolus (t) 14:59, 15 September 2023 (UTC)

The stable version of the article has the big issue of not having a section "Definition". I corrected this issue in this version], which has been reverted by 慈居. This is this version that must be compared with 慈居's draft. There are two main diferences between the two versions, the definition of intervals and the section on posets and preorders.

Definition of intervals: 慈居 replaces the conceptual definition an interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset with a by-case definition (a different definition for each possible case for the end points (thus 9 definitions). I strongly opposes this, since it is confusing for non specialists. Moreover the definitions of the different cases is clearer if they are presented as possible properties of intervals, as in the stable version. (in both versions there is a large overlap between the definitions of open, closed, left-open, etc., and the section of the classification, which needs to be fixed). IMO, 慈居's change of the definition is definitively not an improvement.

Section on posets and preorder: 慈居 introduces the concept of "order convexity", which is not a well established terminology; otherwise, it would already be defined in Wikipedia. It results that his definition of intervals contradicts the usual one. In particular, the draft asserts that the rational numbers whose square are less than 2 is not an interval in $\mathbb {Q} .$ I do not oppose to have a section on intervals in preordered sets, but, firstly, it must be reliably sourced (a book on mathematics for economy is not sufficient in a section presented as pure mathematics); secondly, if there are reliable sources which assert that the rational numbers whose square are less than 2 is not an interval, it must be explained why, and in which context this change of the usual terminology is useful.

In summary, my version of the definition is (in my opinion) better than 慈居's one, and the section on preorder requires much further work (sourcing and context explanation) to be acceptable. D.Lazard (talk) 16:51, 15 September 2023 (UTC)

I have no personal interest in doing it, but if I were going to make this kind of change I would try to do a more extensive literature survey of the use of the term "interval" with respect to posets/etc., starting with the most popular survey books in order theory, so that any definition in Wikipedia covers the most common definition(s) and optionally discusses less common ones. For instance, there are a number of relevant looking papers at google scholar. –jacobolus (t) 19:46, 15 September 2023 (UTC)

I can do a further investigation about the usage of the related terms for those papers accessible to me, but I'll feel upset if you assume that I have never done any. Some time before my edits in Engliwh Wikipedia, I have already applied my edits to Korean Wikipedia (see ko:구간), and since then I have searched for and read plenty of papers and books mentioning intervals and convex sets. The treatment is quite unanimous throughout different authors as to what they mean. The inconsistency lies in which intervals are "closed intervals", "open intervals", "rays", etc. But the intervals and the convex sets are never interchanged, as far as I see. 慈居 (talk) 22:49, 15 September 2023 (UTC)

@D.Lazard Inre your version, I think I would make an explicit note that for a partially ordered set not all of the elements of the interval need to be comparable to each-other, only to the endpoints. –jacobolus (t) 21:26, 15 September 2023 (UTC)

In my version, I have simply reproduced the definition of the main article. I agree that more explanations would be useful, but this cannot be done while it is unclear whether this section should be generalized to preorders. D.Lazard (talk) 14:18, 16 September 2023 (UTC)

Incomparability also arises in partial orders. 慈居 (talk) 21:23, 16 September 2023 (UTC)

The stable version of the article has the big issue of not having a section "Definition". I corrected this issue in this version], which has been reverted by 慈居.. That's an incomplete description of what has happended. I have created the section Definition before you do. You removed it, and renamed a section called Terminology as Definition.

[...] a subset of the real numbers that contains all real numbers lying between any two numbers of the subset [...] I have a dejavu while I'm explaining this, but this is the definition of (order-)convex sets which is equivalent to intervals in the real line but weaker in general. See Heath's and Steen's papers in User:慈居/sandbox#References.

[...] a by-case definition (a different definition for each possible case for the end points (thus 9 definitions). I strongly opposes this, since it is confusing for non specialists. Yes, I have missed and have to complete the conceptual definition that an interval is a subset that has one of the aforementioned nine forms, since this may not be clear for the general readers.

慈居 introduces the concept of "order convexity", which is not a well established terminology Convex sets (and order-convex sets which I used to distinguish it from convex sets in (real topological) vector spaces) are well established terminologies. See Heath's and Steen's papers in User:慈居/sandbox#References. Both authors call it convex sets. For the alternative terminology of "order-convex sets" which was easy to google out, see e.g. [4] and [5].

a book on mathematics for economy is not sufficient in a section presented as pure mathematics This is a subjective underrating of the reference. Economy uses mathematics. The nobel prize winning Arrow's impossibility theorem invovles the use of preorders (although the English Wikipedia article restricts to the case of partial orders). Vind's textbook is a reliable source, even for the purpose of mathematics.

secondly, if there are reliable sources which assert that the rational numbers whose square are less than 2 is not an interval, it must be explained why, and in which context this change of the usual terminology is useful. This serves as a counterexample supporting the following referenced sentence,

Clearly every interval in $X$ is convex but not conversely, […]

— Heath's paper in User:慈居/sandbox#References

and as an answer to the following exercise in Munkres.

7. Let $X$ be an ordered set. If $Y$ is a proper subset of $X$ that is convex in $X$, does it follow that $Y$ is an interval or a ray in $X$?

— Munkres's textbook in User:慈居/sandbox#References

I don't have a reference for this specific counterexample yet, but it naturally arises in the context of calculus when constructing the real numbers as the Dedekind cuts of rational numbers. $\{x\in \mathbb {Q} \colon x^{2}<2\}\cup \mathbb {Q} ^{-}$ is exactly the dedekind cut that represents the square root of 2. It is a lower set, which is a special class of order-convex sets, and not an interval because it is not of one of the nine forms (with rational or infinite endpoints).

[…] and the section on preorder requires much further work (sourcing and context explanation) to be acceptable. You are no referee. The discussion is to decide whether to keep my recent edits. It should depend on whether my edits violate existing policies or guidelines, not on whether they meet some personal fussy criterions.

慈居 (talk) 21:36, 15 September 2023 (UTC)

The discussion is to decide whether to keep my recent edits: No, Wikipedia is not a collection of individual work, but a collaborative work. So the discussion is not only to decide whether to keep your recent edits, but, if kept, how it should be edited by you or by others to meet Wikipedia's quality standards. D.Lazard (talk) 08:44, 17 September 2023 (UTC)

That will be more convincing if you have said anything about what if it's kept. 慈居 (talk) 09:32, 17 September 2023 (UTC)

I'm replying to say that it's just not true that undue weight could only occur if it is an incorrect or deprecated theory. ByVarying | talk 21:56, 15 September 2023 (UTC)

This section is too long for remaining useful. Nevertheless, it seems to have reached a consensus on several points that I'll summarize below. So, I suggest to close it, and to discuss the points for which there remain disagreements in separate new sections.

• Overall structure: Three versions have been proposed for the article, the current one, my version, and 慈居's draft. They have essentially the same structure; so, I consider that a consensus is reached for the general structure. Moreover, there is no objection against my version, and two (myself an another editor) of the four intervening editors support it explicitly. So, I'll restore it as a basis for future improvements of the article.
• Intervals in posets and preorders: There are several different points for which there is no consensus.
  • Should this be a section or a subsection of § Generalizations? IMO, this is a minor point, which will be easier to solve, when the other points will be solved.
  • As the generalization from posets to preorders is mathematically trivial, one has the choice between either to present the case of posets and then to say that this applies also to preorders, or to present directly the case preorders, and to say that partial and total orders are preorders. I am in favor of the latter choice, per WP:TECHNICAL, but this deserves a specific discussion.
  • Terminology and content fork: This is the main point of disagreement. I'll soon open a thread about this.

Please, stop editing this section (except to closing it formally). If you disagree with my conclusions, please, express your disagreement in a new section. D.Lazard (talk) 10:19, 17 September 2023 (UTC)

I totally disagree with your interpretation of the current status of the consensus. I'll explain in a new section as you request. 慈居 (talk) 10:58, 17 September 2023 (UTC)

References

Current status of the consensus

I prefer my version compared to yours. @Jacobolus: seem to have made a positive comment about my version. In my opinion, it was a little hasty to restore your preferred version, because a consensus still forming. 慈居 (talk) 11:09, 17 September 2023 (UTC)

What I said is that I think your later version was better than your earlier version that was the subject of the initial edit war, because it deferred the more technical topic to further down the page. I tend to agree with D.Lazard that the version with all of the explicit cases is a bit harder to make sense of as the primary definition of interval at the top of the page. I haven't ever thought about intervals in general posets so don't feel qualified to judge how that topic would best be presented. –jacobolus (t) 15:37, 17 September 2023 (UTC)

The problem is that the primary definition doesn't work well in the poset setting. For example, any antichain will be an interval in that definition. We also have to consider the established term "convex set" and and all the derived terms such as "convex component", for the primary definition. The new definition is apprantly much longer, but it is an unambiguous definition. (An interval is a subset $I$ such that $I$ have one of the nine forms with respect to endpoints $a,b\in \mathbb {R} \cup \{-\infty ,\infty \}$.) 慈居 (talk) 04:54, 18 September 2023 (UTC)

I disagree for the same reasons I originally commented about. There's no reason why the principal definition this article gives for the real case needs to directly generalize to other (poset) cases. ByVarying | talk 12:40, 18 September 2023 (UTC)

Why not? This is apprantly second to which is the actual standard definition, but the primary definition is not even a standard one, as seen e.g. in the reference below. And this is my personal experience, but at high school, I was only taught the endpoint definition. I wonder if this is the same for you, too. 慈居 (talk) 16:49, 18 September 2023 (UTC)

As a reference of the "by-case" definition, I quote a definition in a calculus textbook by the great mathematician Terrence Tao.

Definition 9.1.1 (Intervals). Let $a,b\in \mathbb {R} ^{*}$ be extended real numbers. We define the closed interval $[a,b]$ by

$[a,b]:=\{x\in \mathbb {R} ^{*}\colon a\leq x\leq b\},$

the half-open intervals $[a,b)$ and $(a,b]$ by

$[a,b):=\{x\in \mathbb {R} ^{*}\colon a\leq x\leq b\};\qquad (a,b]:=\{x\in \mathbb {R} ^{*}\colon a<x\leq b\}$

and the open intervals $(a,b)$ by

$(a,b):=\{x\in \mathbb {R} ^{*}\colon a<x<b\}$

We call $a$ the left endpoint of these intervals, and $b$ the right endpoint

— page 212, Definition 9.1.1

The author explains in a lemma what is meant by "a subset is an interval".

Lemma 9.1.21. Let $I$ be an interval (possibly infinite), i.e., I is a set of the form $(a,b)$, $(a,b]$, $[a,b)$, $[a,b]$, $(a,+\infty )$, $[a,+\infty )$, $(-\infty ,a)$, or $(-\infty ,a]$, with $a<b$ in the first four cases. Then every element of $I$ is a limit point of $I$.

— page 215, Lemma 9.1.21

慈居 (talk) 05:52, 18 September 2023 (UTC)

References

Intervals in posets and preorders

Following Jacobolus' suggestion, I'll take some time to further investigate and make a survey about the topics of intervals in posets and preordered sets. But before that, I have noticed a good (although I hate to say it) idea suggested by D.Lazard to write for posets and make a short note that it applies also to preordered sets. That's what I can accept, provided that It'll be clear from context that the formuation in the preorder setting is parallel to that for posets. And I can handle it. 慈居 (talk) 11:18, 17 September 2023 (UTC)

Section "Topological algebra"

Section § Topological algebra is blatant original reasearch. It is about the direct product (as rings) of the field of the real number with itself. This ring is well known and considered in several Wikipedia articles, but has no relation with intervals, except that its elements and intervals are both defined by ordered pairs of real numbers.

So, I'll remove this section as soon as the above dispute will be resolved. D.Lazard (talk) 09:42, 14 September 2023 (UTC)

You're right, unfortunately. The references provided in the section is about the interval analysis. By definition, the interval analysis $I(\mathbb {R} )$ of the real numbers (the reference also mentions $I(\mathbb {R} ^{n})$ and $I(\mathbb {C} )$, if you wanna know) has the same addition as $\mathbb {R} \times \mathbb {R}$ (if one identifies each $[a,b]$ as a pair of end points), but the multiplication is

$[a,b]\cdot [c,d]=[\min\{ac,ad,bc,bd\},\min\{ac,ad,bc,bd\}].$

They are different matters. 慈居 (talk) 09:51, 15 September 2023 (UTC)

Interval analysis is obviously a topic worthy of noting and I'm considering rewriting the section using the correct definition. See User:慈居/sandbox for future updates. 慈居 (talk) 09:51, 17 September 2023 (UTC)

Terminology and content fork

In User:慈居/sandbox#Intervals in posets and preordered sets, user 慈居 proposes a new section for Interval (mathematics). This a WP:REDUNDANTFORK of Partially ordered set#Intervals except for two points:

• The trivial generalization from posets to preorders.
• A change of terminology: intervals in the article are called "order-convex" in the draft, and intervals in the draft are called "intervals that can be represented in interval notation" in the article.

The first point is easy to fix, as it suffices to say (in Interval (mathematics) and Partially ordered set#Intervals that the definitions apply directly to preorders.

For the change of terminology, it must be clearly established which terminology is used in reliable secondary sources. Partially ordered set#Intervals is not explicitly sourced, and the only secondary source produced by 慈居 is a book published in a Studies in Economic Theory series, which is not sufficient for establishing the terminology in general mathematics.

My impression is that the most common terminology is that of Partially ordered set#Intervals. I may be wrong, but, as the terminology of this article is here since a long time, it must not be changed unless a strong evidence is provided that this is not the most common terminology.

In any case, it must be mentioned that both terminologies are used, and, if a terminology is used only in a specific context, this must be made explicit.

Independently from this question of terminology, the two sections must be merged either here, with a summary in Partially ordered set#Intervals or in the latter article with a summary here. Personally, I prefer the latter solution, as it is the easiest to implement.

My personal opinion on the terminology is that "interval" must be reserved to totally ordered sets, and that "order-convex" should be used in more general cases, as being less confising. However my personal opinion does not matter, as this is the common usage that prevails. D.Lazard (talk) 11:28, 17 September 2023 (UTC)

For the change of terminology, it must be clearly established which terminology is used in reliable secondary sources. Partially ordered set#Intervals is not explicitly sourced, and the only secondary source produced by 慈居 is a book published in a Studies in Economic Theory series, which is not sufficient for establishing the terminology in general mathematics. What about Heath's paper and Steen's paper (on general topology) in User:慈居/sandbox#References? What about [6] and [7]? 慈居 (talk) 11:51, 17 September 2023 (UTC)

What about you? Do you have any references that supports your view in terminology? 慈居 (talk) 12:00, 17 September 2023 (UTC)

Again, my personal view on terminlogy does not matter, but Wikipedia must be coherent with itself (this would not be the case if your draft would be moved without change in the main space), and must follows the most common usage. For establishing the most common usage, research papers are not really useful, as it is common that authors provide their own definitions if they are more convenient for their particular usage. So, this is textbooks on order theory, and especially the most widely used ones, that must be used. In particular, the definitions given in references 1 to 4 of your reference 6 must be checked for the definition of intervals (reference 1 is not a textbook, but I believe that it is written by the founders of the theory; so, it must be considered). I have no easy access to these references; please, provide an quotation of the definition given there. D.Lazard (talk) 13:42, 17 September 2023 (UTC)

For clarity, which reference do you mean by "reference 1~6"? Can you mention the author(s) or title(s) of the reference? The numberings might change as I update the article. 慈居 (talk) 13:50, 17 September 2023 (UTC)

I mean the references 1 to 4 in [6]. D.Lazard (talk) 13:57, 17 September 2023 (UTC)

For intervals and convex sets in posets, I also have this reference, which is a math textbook as you insist, which quotes (p. 10, Definition)

Definition Let $(P,\leq )$ be a poset. For $a,b\in P$ the closed interval from $a$ to $b$ is the set $$[a,b]=\{x\in P|a\leq x\leq b\}$$ The initial segment up to $a$ is the set $$(-\infty ,a)=\{x\in P|x<a\}$$ and the closed initial segment up to $a$ is the set $$(-\infty ,a]=\{x\in P|x\leq a\}$$ A nonempty subset $S$ of a poset $P$ is convex if whenever $a,b\in S$, then $[a,b]\subseteq S$.□

慈居 (talk) 12:34, 17 September 2023 (UTC)

This shows that "order-convex" is an established terminology, but does not provide a general definition of intervals. In particular, it is unclear whether an "initial segment" is an interval. D.Lazard (talk) 13:53, 17 September 2023 (UTC)

Well, are you interested in what the economy textbook says? The context is highly mathematical and clearly defines what the intervals (in preorders) are. 慈居 (talk) 14:03, 17 September 2023 (UTC)

Apart from that, I don't have somehting else yet. I'm sure I can find another reference but It'll take some time. 慈居 (talk) 14:05, 17 September 2023 (UTC)

Independently from this question of terminology, the two sections must be merged either here, with a summary in Partially ordered set#Intervals or in the latter article with a summary here. Personally, I prefer the latter solution, as it is the easiest to implement. – there is a third solution which is to make a new article about this topic linked from both. That would only make sense if there's enough to say about the specific topic that it would spill out of the bounds of a short summary section at either existing page. –jacobolus (t) 15:42, 17 September 2023 (UTC)

I don't know, this concept seems pretty out of the way and it's not clear to me from the sources that have been given that it's useful or notable in its own right. I think it should not have its own page unless more important sources are found. ByVarying | talk 00:59, 18 September 2023 (UTC)

Do you think that the contents in Partially ordered set#Intervals should also be deleted then? 慈居 (talk) 03:23, 18 September 2023 (UTC)

No, as I said, I think this topic should not have its own article. It's fine to cover it in a section of a different article. ByVarying | talk 03:34, 18 September 2023 (UTC)

Isn't interval (mathematics) a best fit for that article? why should they be covered in a much broader topic like partially ordered set? 慈居 (talk) 03:41, 18 September 2023 (UTC)

Do you think I'm saying the article Interval (mathematics) shouldn't exist? I'm not; I'm saying there shouldn't be a stand-alone article about intervals in posets and preorders. Nowhere did I say that the coverage of those topics should be in this article or at the poset article, either. ByVarying | talk 04:31, 18 September 2023 (UTC)

No, I don't think you're saying that. What I think is that, if as you say, the topic should be in a section of a different article, than the best of such article is interval (mathematics). 慈居 (talk) 04:50, 18 September 2023 (UTC)

I think this is a good idea, but I'd rather leave this to other editors. Improving based on my edits may not yield a best result. And it will be good to have different treatments in different styles and phrasings of the same topic. Editors can of course move my edits to a separate article if they want to, but I won't do that myself. 慈居 (talk) 03:48, 18 September 2023 (UTC)

Let me quote the Vind's textbook anyway. It clearly defines what intervals are and is in the most general context of preordered sets.

Notation 1 A preordered set will be denoted $(S,\succsim )$. An ordered set will be denoted $(S,\succeq )$.

— page 10, Notation 1

Definition 11 The set

$\{s\in S|b\succsim s\succsim a\}$ is denoted $[a,b]$

and is called the closed interval with $a$ as the initial point and $b$ as the terminal point.

${\mathopen {]}}a,b{\mathclose {[}}=\{s\in S|b\succ s\succ a\}$

is called an open interval. We shall also call

${\mathopen {]}}a,b{\mathclose {]}}=\{s\in S|b\succsim s\succ a\}$ ${\mathopen {[}}a,b{\mathclose {[}}=\{s\in S|b\succ s\succsim a\}$ ${\mathopen {[}}a,{\rightarrow }{\mathclose {[}}=\{s\in S|s\succsim a\}$ ${\mathopen {]}}a,{\rightarrow }{\mathclose {[}}=\{s\in S|s\succ a\}$ ${\mathopen {]}}{\leftarrow },a{\mathclose {[}}=\{s\in S|a\succsim s\succ a\}$ ${\mathopen {]}}{\leftarrow },a{\mathclose {[}}=\{s\in S|a\succ s\}$

intervals.

— page 11, Definition 11

— Preceding unsigned comment added by 慈居 (talk • contribs) 06:18, 18 September 2023 (UTC)

References

At the present stage, I am convinced that the term of (order)-convexity is well established as well as the use of "interval" only for order-convex sets that have endpoints. Still open is the question whether the other terminology (given in Poset) is also in common use in the context of order theory. I'll edit Poset#Intervals for mentioning both terminologies. If it appears that the current terminology of this section is uncommon, the section should be further edited.

Also, at the present stage, it seems that there is consensus that the topic of intervals in order theory should be developed in a single article, with summaries in the other articles, but there is no consensus on this single article. My opinion is that this should be in an article that is mainly about order theory. For the moment, the topic of intervals in order theory consists mainly of definitions and trivial theorems. So, I agree with ByVarying that it is not enough for a standalone article. Nevertheless, if more relevant content is added, a standalone article could be useful.

For preparing this, and for an easier navigation, I'll create the redirects Interval (order theory), Order convexity, Order-convex, Convex set (order theory). For the moment, their common target will be Partially ordered set#Intervals; the target will be easy to change if the articles structure will eventially be changed. D.Lazard (talk) 14:38, 18 September 2023 (UTC)

I'm glad to hear that. It sounds a great idea to have a separate article for these topics. However, I think my edits still can serve as a summary to that. If a summary is acceptable, why not let it be more than just one or two lines? The new article can be as long as the whole interval (mathematics), and can be presented way better than mine. 慈居 (talk) 17:05, 18 September 2023 (UTC)

what's the point of intervals

This article is currently almost entirely focused on (frankly not very interesting for most readers) pedantic trivia about formal definitions of intervals. While that material is probably worth including, it would be useful to expand the article with (much!) more information about how/why intervals are used and how they relate to other subjects.

We have a mention of interval arithmetic in the lead section but no section here summarizing it.

We mention the epsilon-delta definition of continuity as another throwaway line in the lead, but don't discuss it in the article body.

We make no mention of the cardinality of a real interval. The linear continuum is mentioned as a kind of weird non sequitur in a section that doesn't make any sense to me, but there's no discussion of real intervals as representing a continuum. Line segment is only listed as a "see also".

We only describe constructing unions and intersections of intervals as a pair of throwaway sentences but don't explain why anyone wants to do that or what the result is of taking the union of intervals that don't overlap. We should probably discuss both finite and infinite unions and intersections of intervals here, and what kind of sets they can form.

We don't talk the linear functions mapping any finite interval to any another (given matching endpoint types), which are used constantly.

We don't describe the standard intervals $[0,1]$ and $[-1,1],$ or mention linear interpolation. We don't talk about the way a parametric curve is the image of an interval.

We don't talk about reparametrizing intervals, e.g. by fractional linear transformations.

We don't describe in general functions of an interval, or ways that those functions can be approximated (e.g. by Chebyshev polynomials).

We don't talk about piecewise functions with separate definitions on adjacent intervals (and sometimes separate definitions at isolated endpoints). We don't talk about spline interpolation.

We never mention the concept of a periodic interval (red link!!) or explain its relationship to the ordinary kind.

Etc. etc. –jacobolus (t) 19:04, 19 September 2023 (UTC)

I'm glad that we have here someone with these insights. It seems a great idea to have those (apprantly much more interesting) topics covered. Many of what you have mentioned is beyond my current knowledge and it will be delighting to see them covered by other volunteers (e.g. you!). Maybe I can say something about the role of products of unit intervals $[0,1]$ in general topology as universal spaces for Tychonoff spaces and compact Hausdorff spaces, when I get rid of my laziness. 慈居 (talk) 19:20, 19 September 2023 (UTC)

Sorry, I'm not trying to criticize authors of the page as it is. All I mean is: intervals are a basic building block used constantly all over mathematics. To strain a metaphor, right now we have an article entirely focused on the shape, texture, and weight of the block, but barely any mention whatsoever of what someone might want to build with it. –jacobolus (t) 19:26, 19 September 2023 (UTC)

I think it is a good criticism to the current status of the page, so please don't worry. I agree with the message behind your metaphor though I haven't thought about this before. 慈居 (talk) 19:40, 19 September 2023 (UTC)

I completely agree with you (Jacobulus). IMO, before expanding the article in the directions that you suggest, we have first to clean up, simplify and correct the trivia. I did several tentatives in this direction. Some are still there (despite several reverts), such as the ubiquity of intervals in analysis, or the short description in the lead of interval arithmetic. But most are changed for restoring or adding pedantry, or even for restoring errors of previous versions.

For example, before my today edits, there was intervals that were closed sets without being closed intervals! I just see that this inconsistency has been restored.

Similarly, the first sentence In mathematics, a (real) interval is a set of real numbers without "hole", in the sense that, with any two numbers, it contains all real numbers between them has been removed (or moved down) several times, for putting at the top a technical definition that is grammatically/logically incorrect ("between" excludes the endpoints).

So, I give up to improve the article unless 慈居 will stop definitively to edit this article. As he seems to not understand that his behavior is WP:disruptive editing, it is maybe time to open a thread at WP:ANI. D.Lazard (talk) 21:13, 19 September 2023 (UTC)

This seems like an unnecessarily aggressive summary. –jacobolus (t) 22:05, 19 September 2023 (UTC)

Is this relevant to the topic of this section? You don't seem to see what I have done to seek compromise, e.g. rephrasing the definition upon your claim that the definition is too technical, removing Template:Rp when you say "no reference in the lead", etc. If you think that I'm only a disruptive editor and want to draw administrators' attention, go on, who can stop you. 慈居 (talk) 04:10, 20 September 2023 (UTC)

Note on the definition based on convexity

There are currently two equivalent definitions of the notion of intervals (I will rephrase them to make them unambiguous).

• Via convexity. An interval is a subset $S$ of the real numbers such that whenever $x,y\in S$ and $x\leq z\leq y$ we have $z\in S$
• Via interval notation. An interval is a set $S$ such that either of the following hold for some real numbers $a,b$.
  • $S=(a,b)$
  • $S=[a,b]$
  • $S=(a,b]$
  • $S=[a,b)$
  • $S=(a,+\infty )$
  • $S=[a,+\infty )$
  • $S=(-\infty ,a)$
  • $S=(-\infty ,a]$
  • $S=(-\infty ,+\infty )$

The latter definition is added in the lede by me and rephrased by Jacobololus to be less technical. Now I'd like to argue here that, the former definition is not an easy one as it seems. First of all, the convexity definition requires the knowledge of supremum and infimum to find the endpoints, which is usually taught only in (advanced) calculus courses. The latter definition is acceptable even to people with a basic intuition of real numbers and their order relations, and the endpoints are already there since they are predetermined. Second, general audience understand a concrete definition better than an axiomatic one. That's why we don't start the first-year university math course by category theory. Non-specialist readers might struggle to figure out that the convexity axiom characterizes line segments or rays in the real line. Finally, the latter definition seems the only one that is used in high school mathematics. I have verified that this is the case in South Korea and China by checking some textbooks there and further verification will be appreciated. 慈居 (talk) 06:29, 20 September 2023 (UTC)

In my opinion you are overthinking it. If you want a more technical definition, the clearest is probably to say that a real interval is a connected set of the real numbers (e.g. this is the definition adopted in Hickey, Timothy; Ju, Qun; Van Emden, Maarten H. (2001). "Interval arithmetic: From principles to implementation" (PDF). Journal of the ACM. 48 (5): 1038–1068.) since the concept of connectedness is pretty intuitive and close to the plain English meaning of the word, so therefore less technical readers who don't know the precise meaning of connected as topology jargon won't be misled. But there are obviously many equivalent definitions you might choose. The lead section in particular should not be picking the ideal definition based on deep technical arguments. Instead the goal should be to convey the concept in a concise and clear way for a general audience – imagine someone with a high school level education who is curious but also busy. If in the lead section you either (a) throw in lots of jargon and advanced concepts, or (b) flood the prose with formulas, you will spoil the article's legibility for less technical readers. –jacobolus (t) 08:02, 20 September 2023 (UTC)

One example of the "convexity" definition can be found in Royden & Fitzpatrick's Real Analysis:

Given real numbers a and b for which a < b, we define (a, b) = {x | a < x < b}, and say a point in (a, b) lies between a and b. We call a nonempty set I of real numbers an interval provided for any two points in I, all the points that lie between these points also belong to I. Of course, the set (a, b) is an interval, as are the following sets: [a, b] = {x | a ≤ x ≤ b}; [a, b) = {x | a ≤ x < b}; (a, b] = {x | a < x ≤ b}.

This seems pretty clear and explicit, though probably needs slightly more elaboration for a Wikipedia audience. (This kind of formal definition should not go in the lead section.) –jacobolus (t) 08:54, 20 September 2023 (UTC)

I agree that we should avoid technical arguments when possible (especially in the lead section). I don't mean to apply the definition with so many formulae to the article as I present in this section (I present this way here just to give an unambiguous definitions, to the definitions). Every definition other than the endpoint one unavoidably needs to be justified that they are exactly the segments and rays in the real line. And this requires at least a little bit of order theory in the real line. For example as you may alreay know, If $S$ is a connected subset of the real line, then it is convex by an argument that involves the definition of connectedness (if $x,y\in S$, $z\in \mathbb {R} \setminus S$ and $x<z<y$, then $S$ is splitted by disjoint open sets $(-\infty ,z)$ and $(z,+\infty )$.) Then we have to take $a=\inf S$ and $b=\sup S$ and argue that, if e.g. both $a,b$ are real, $(a,b)\subseteq S\subseteq [a,b]$. It will be good to have a definition using intuitive terminologies, but not at the cost of increasing internal difficulty. 慈居 (talk) 08:57, 20 September 2023 (UTC)

I think you are interposing "internal difficulty" and need for "justification" based on your external knowledge rather than any proposed content or requirements for this article. Wikipedia does not need to prove or justify every result, only to explain and to be based on reliable sources. Someone wanting a detailed analysis of "order theory in the real line" should look for it elsewhere. Even if such a thing were required in the context of a journal paper in pure mathematics, it can be safely skipped here. –jacobolus (t) 09:06, 20 September 2023 (UTC)

I mean, even an explanation will tend to mention advanced topics (like least upper bound property in the current article), and this explanation can be placed behind basic materials if the convexity definition is a secondary definition. 慈居 (talk) 13:44, 20 September 2023 (UTC)

In your quote of Royden & Fitzpatrick, it is only said that $(a,b)$ etc. are intervals, and that's an easy part. What is difficult (for general audience) is to show that every intervals (in the sense of convexity) are of those forms. 慈居 (talk) 09:00, 20 September 2023 (UTC)

All this article needs to do is state that intervals have these forms and link to a source. This does not need to be explicitly proven inline in the text. –jacobolus (t) 09:09, 20 September 2023 (UTC)

Well, I think a proof or a detail also deserves a seat here. External sources are often paywalled or not easily accessible otherwise. And that's one of my motivations to contribute here. 慈居 (talk) 09:29, 20 September 2023 (UTC)

Whether external sources are paywalled is not really Wikipedia's concern. Readers who care enough about subtle/advanced technical details (say, math grad students) are likely to be able to find access to paywalled sources. But I'm also sure a publicly accessible source can be found for nearly any claim this article would ever want to make, if anyone strongly cares about that. If you want to write up a detailed presentation with full proofs of every claim you are welcome to publish such a thing on arXiv or a blog post or a paper in some open-access journal. It just starts to feel like an out-of-scope digression here. –jacobolus (t) 16:37, 20 September 2023 (UTC)

I think that rationale goes against the principle that Wikipedia is not a textbook. ByVarying | talk 11:51, 20 September 2023 (UTC)

No one says that it is. Taking wp:technical into consideration does not imply that Wikipedia is a textbook. 慈居 (talk) 13:10, 20 September 2023 (UTC)

Yes, but the convexity-based definition itself is not too technical at all. You're claiming it's too technical because one needs to have specific knowledge in order to derive certain facts from it but that's not what wp:technical is about. If that were the true interpretation of wp:technical then every science article on this website would be tens of thousands of words long. ByVarying | talk 15:22, 20 September 2023 (UTC)

I should have posed this differently. I should have noticed this earlier but my concern is some kind of overall difficulty. For example, the convexity definition leads to the supremum-infimum definition of the endpoints, while the notational definition of intervals leads to the predefined notion of the endpoints. So in my opinion, it's either technical here, or technical there. The difficulty of the definition of intervals comes from its number of cases. The difficulty of the definition of endpoints comes from the sup-inf notions. I was pretty sure that it's better to accept the former one (and by accept I only mean to put it first), but now I hesitate as this is opposed by at least three people. 慈居 (talk) 19:40, 20 September 2023 (UTC)

Just in case, by notational definition I don't always mean a definition using a bunch of formulae. In the lead section a symbol-free presentation will be enough, as it is now. 慈居 (talk) 20:03, 20 September 2023 (UTC)

It is not necessary to start with a discussion of endpoints as supremum/infimum. This can be mentioned further down the page among the properties of intervals, but it's distracting to belabor the point. –jacobolus (t) 21:18, 20 September 2023 (UTC)

Of course we can define intervals as convex sets first, explain its equivalence with the other definition, define endpoints from there and say later that endpoints are suprema and infima. That's fine but in my opinion, it's kind of a weird structure. Usually a definition is responsible for all derived notions, i.e. they are all defined relying on the definition, and then another definition is ready to be introduced. It might seem "belaboring the point" to you, but I have to somehow deliver my point and it would less feel like so if this is my first language. 慈居 (talk) 05:19, 21 September 2023 (UTC)