Talk:Initial segment

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@David Eppstein: I understand there are some overlaps but a discussion like an initial segment as a map between coalgebras or a role in recursion just doesn’t fit natural into an article like upper set. Such a material would look very out of place there. Taku (talk) 23:55, 11 April 2026 (UTC) Another problem for me was: the upper set article focuses on upper sets. I know facts or properties for upper sets would also hold for lower sets by reversing the order. But psychologically and conceptually, that style isn’t natural: that would be like not having an article on filters if we have an article on ideals or just having an article on an inductive limit and not on a projective limit. Taku (talk) 00:28, 12 April 2026 (UTC)Reply

That argument would make sense only if your article was about coalgebras or recursion. Instead there is zero content about those things and an article on a topic that we already have an article on, the poset definition. The fact that the upper set article has that title is irrelevant; it is about both upper and lower sets.

Refocus this article on something other than poset lower sets or expect to defend it against a deletion nomination. Your choice. —David Eppstein (talk) 01:38, 12 April 2026 (UTC)Reply

I said "some" overlaps because there are already some differences. For example, this article already has more stuff in the case of well-ordered sets, while the upper set article has essentially none (as far as I can tell) and I’m not sure, for things like, for example, an ordering of well-ordered sets by initial segments, the upper article is a good place for that. I didn’t want to make the upper article too oriented toward set theory and that’s why I thought a separate article is a better idea. I’m certainly not objecting to having an article on both upper and lower sets, as they are related, but in set theory or proof theory, an upper set just isn’t used like initial segments are used; so the upper article is structurally problematic for those needs. Like filter vs ideal I mentioned, a context not just a mathematical definition matters in deciding to have a separate article or not. Taku (talk) 03:24, 12 April 2026 (UTC)Reply

You keep talking about other areas of mathematics, previously recursion theory and algebra, now set theory and proof theory. If you want an article about those other areas, write an algorithm about those other areas. Don't make it about posets, because we already have an article on this topic for posets. If our poset article is missing some content that would be relevant for applications of its concepts, then that is the place it should be included. Don't make a WP:CONTENTFORK on exactly the same topic. —David Eppstein (talk) 04:12, 12 April 2026 (UTC)Reply

An example of why contexts matter would be: having articles on both vector space and module. Your argument there would be: because a vector space is just a module, we shouldn’t have an article on a vector space. But that’s not *natural* right? Because they appear in *different contexts*, linear algebra and abstract algebra, respectively. The way to see how contexts differ in our discussion would be looking at references we are using. This article uses those in the foundation of mathematics like Halmos' native set theory. Such references would be weird in an article on a topic in order theory. It's not about applications but rather contexts: an order-theory notion appears in different contexts. I think the difference is important enough for separate treatment. (By the way, by algebra, I was thinking of a category-theoretic usage in foundations. This is probably an unfortunate terminology.) Taku (talk) 07:58, 12 April 2026 (UTC)Reply

A vector space is over a field, a module is over a ring. But here, we have: a lower set is over a poset, an initial segment is over a poset. They are exactly the same thing. —David Eppstein (talk) 16:57, 12 April 2026 (UTC)Reply

As I said above, the article mostly considers an initial segment of a well-ordered set. I stated the definition for a poset in general since the notion makes sense more generally. So, the analogy actually works: a vector space is just a module but when the coefficient ring is a field. Similarly, the article considers an initial segment mostly when the poset is well-ordered. (Thus, actually I was also thinking of putting materials here at well-ordered but that article seems more about a particular ordering. ) In fact, the terminology difference is already telling: the term "lower set" is almost never used in the context of the article; so it's more like the same concept happens to be used in different contexts. Also,while a well-ordered case is the principal case, the article can handle an initial segment in a more general setup (namely, the key is well-foundedness and again that kind of stuff just doesn't belong to the upper article). Taku (talk) 22:11, 12 April 2026 (UTC)Reply

That makes even less sense than before. Well-ordered sets are totally ordered, not just partially ordered. —David Eppstein (talk) 00:50, 13 April 2026 (UTC)Reply

First, I have added a more general definition (the library was closed over the weekend and so I just got a reference on Mon). Apparently, an initial segment is also defined for a binary relation as well; this definition is already not in the upper article. As for well-ordered sets, my point was that a vector space is a module with an additional assumption on the coefficient ring (nonzero elements are invertible). Similarly, here, an initial segment is considered with the additional condition that the ordering is well-order (each nonempty subset has a least element). So, the analogy does work. Anyway, since you're clearly not convinced, I think we should seek a third opinion. I will ask the math project members for inputs (after your response). P.S. I have added a hatnote to clarify the scope. Taku (talk) 08:11, 13 April 2026 (UTC)Reply

Yes, I am not convinced. A hatnote is not enough. Change the lead sentence to something other than being about lower sets of posets or restore the redirect to our article on lower sets of posets. —David Eppstein (talk) 17:39, 13 April 2026 (UTC)Reply

I agree with David that, at present, this article is just a fork of upper set. (Similarly we have an article Sine and cosine, Large set (category theory) is a redirect to Small set (category theory) and Small set (combinatorics) is a redirect to Large set (combinatorics), etc.) I could imagine the existence of an article on initial segments of well ordered sets being split off from Upper set but there is not even a full paragraph in this article on that subject at present. The obvious thing to do would be to develop content at Upper set until such a point that the case for splitting becomes clear. --JBL (talk) 23:53, 13 April 2026 (UTC)Reply

Actually most of discussion at the properties section is about initial segments of well-ordered sets; so if we were to do a split, this article is already a result. Also, sine and cosine or large and small sets seem to be arguments *for* a separate article: in set theory, one only considers initial segments. When there is an order-theoretic discussion, the term lower set seems to be preferred in that case (e.g., Taylor in the references), while the term initial segment is reserved for discussion related to induction and decision. So, to repeat, to me, it suggests we are in a situation similar to that of having vector space and modules as separate topics (it's just, in the poset case, initial segments happen to be the same as lower sets). Taku (talk) 08:39, 14 April 2026 (UTC)Reply

most of discussion at the properties section is about initial segments of well-ordered sets Yes; that is the "not even a full paragraph" I referred to. This one paragraph of content will be very happy in a section of Upper set titled "Initial segments of well orders". (And if Module (mathematics) had only one paragraph of content, then that one paragraph would live happily as a section of Vector space titled "Modules over rings".) Generally speaking readers and editors are better off if content about different perspectives on the same (or closely related) objects is housed in single articles, rather than split into microstubs: it makes things easier to find and reduces redundancy. There is nothing about this case that seems exceptional in this regard. --JBL (talk) 18:52, 14 April 2026 (UTC)Reply

Thank you; I didn’t understand what you meant by “full paragraph”. (The properties section has 5 paragraphs and among them, only one doesn’t involve well ordered-ness). While my preference would be a separate article, I can also see your preference to have a single article coveting different perspectives (my view is that approach is not optimal since that would be confusing; e.g., the term lower set isn’t used as much as initial segment in set theory. Also, a shorter article is easier to read. But anyway this is a matter of trade-off). Taku (talk) 00:38, 15 April 2026 (UTC)Reply

Since JBL has already given the examples, how about changing the title of the upper set to something like "Sine and cosine"? Splitting an article would require a context like the Isbell duality, but the Yoneda lemma is not split.--SilverMatsu (talk) 02:04, 14 April 2026 (UTC)Reply

Personally, I would redirect upper set to initial segment before I would do the other way round. JRSpriggs (talk) 15:17, 14 April 2026 (UTC)Reply

I opened Talk:Upper set - Requested move 14 April 2026 to discuss this. Elestrophe (talk) 17:11, 14 April 2026 (UTC)Reply

Agreed that the article as written is just a fork of Upper set and should be redirected there. Additionally, there wouldn't be a reason to have an article specifically about initial segments of well-ordered sets; that material belongs in the Well-ordered set article. Elestrophe (talk) 17:02, 14 April 2026 (UTC)Reply

Well (pun intended), initial segments are also considered without total-ordering but just with well founded-ness (a well-ordered set is a well-founded totally ordered set). This was also why I put materials here not in well-order article. Taku (talk) 01:00, 15 April 2026 (UTC)Reply

Thank you all for the inputs. Since the consensus is clear for the merger, I’m going to move the materials in this article to upper set. (I don’t think the redirect would work since most of materials here do not appear there.) —- Taku (talk) 00:43, 15 April 2026 (UTC)Reply

(I will wait at least half a day in case anyone else wants to chime in.) Taku (talk) 00:48, 15 April 2026 (UTC)Reply

I did a fairly crude merge. As should be clear, further editing is needed to make the presentation smoother (of course, at least I will try to do that) Taku (talk) 07:56, 15 April 2026 (UTC)Reply

I made some changes. In particular, you cited the definition of "initial segment" to "Well-ordered sets, Definition 2." of this:

Tao, T (28 January 2009). "245B, Notes 7: Well-ordered sets, ordinals, and Zorn's lemma (optional)". What's new..

(Does a blog by a preeminent scholar count as a WP:Reliable source? It seems not as its a "Self-published source")

However that definition is for well-ordered sets ${\displaystyle X}$ only (not arbitrary posets). It's also more complicated:

a set ${\displaystyle Y\subseteq X}$ such that ${\displaystyle \forall y\in Y,[\min(X),y]\subset Y}$

Moreover I randomly guess that the more-general poset definition is more standard (as that's the one I was familiar with). — Isaac Oscar 11:20, 28 April 2026 (UTC)Reply

@Isaac Oscar: per WP:RSSELF, Self-published expert sources may be considered reliable when produced by an established expert on the subject matter, whose work in the relevant field has previously been published by reliable, independent publications. --JBL (talk) 17:56, 28 April 2026 (UTC)Reply

As JBL already pointed out, Tao should be considered a reliable source. And, yes, ostensibly, Tao uses a bit more complicated definition: in a nutshell, his definition is that an initial segment is a lower set. In practice, this definition is actually more convenient but may be less standard. As the article notes, two definitions are equivalent so it’s really just a matter of presentation. At least, I added a textbook ref for equivalence of two definitions. I can agree maybe we should stick to a standard definition (and so am not going to object to your changes). Taku (talk) 00:55, 29 April 2026 (UTC)Reply

@TakuyaMurata Your changes to the article look good, but what do you mean by a (proper) initial segment? I couldn't find any mention of proper vs. improper initial segments in the citation (my guess would be that a "proper" initial segment of a poset ${\displaystyle X}$ is meant to be an initial segment not equal to ${\displaystyle X}$ itself, but that should be all initial segments as a poset is always irreflexive, i.e. ${\displaystyle x\notin \{y\in X|y<x\}\subset X}$, since ${\displaystyle x\not <x}$) — Isaac Oscar 11:56, 29 April 2026 (UTC)Reply

Yes, by non-proper or improper initial segment, I meant an initial segment that is X. X itself isn’t an initial segment determined by an any element. I thought it’s important to mention X itself counts as an initial segment (If X is not an initial segment, the union of initial segments may not be an initial segment, etc.) Taku (talk) 00:29, 30 April 2026 (UTC)Reply