Definition talk:Weak Initial Segment

For reasons of geometric intuition, I think I would prefer to see this term restricted to totally ordered sets (and perhaps even to well-ordered sets), and the term "lower closure" used in other contexts. Does anyone else have an opinion? --Dfeuer (talk) 18:03, 25 February 2013 (UTC)


 * My literature tends to reserve this term (and the strict analogue) to well-orders. So, tentatively agreed, but I'll have to contemplate how to implement this convention; please leave the pages intact until I propose an approach (feel free to post suggestions). In the mean time, good work on restructuring this entangled part of the site. &mdash; Lord_Farin (talk) 18:08, 25 February 2013 (UTC)


 * Thanks. It's still messy and WIPish, which bugs me, but there are a lot of pages to cover. Do you think you could look into the macro matter we were discussing earlier? I don't know nearly enough about the guts of this operation to know how to do that. It doesn't matter so much right now (it's not so hard to search for "\dot\uparrow" and "\bar\uparrow", but once all the strict ones have been switched to the new notation and I want to get rid of the \bars, I'll be putting in something that looks just like the arrows used for logical operators and certain sequence convergence statements (and probably other things I haven't run into yet). --Dfeuer (talk) 18:15, 25 February 2013 (UTC)


 * I have a book that calls $\bar S_a$ the principle lower set of $a$ and writes it as ${\downarrow} \left({a}\right)$. It was made in 1996. --Jshflynn (talk) 18:21, 25 February 2013 (UTC)


 * That alternative term needs to be added to Definition:Lower Closure of Element once that structure is set up (see Definition:Upper Closure). I'll get to it. --Dfeuer (talk) 18:27, 25 February 2013 (UTC)


 * TL; DR: Not presently. Many other stuff to attend to and technical difficulties. Current paradigm (with explanatory lines) works so no immediate need to "fix" it.
 * I'm not presently in favour of a macro system being set up, because I think that in view of site accessibility and comprehensibility we should stick to TeX conventions as much as possible. Should you bring in cogent arguments then I will still defer development of such for a bit because I still have the Help section refactoring and the Logic section rewriting to attend to (although these are a bit halted due to problems with the extension requiring present attention). IOW, it would end up on the big heap immediately, which wouldn't be a good thing. A fluent implementation would also require MathJax and MediaWiki being manually amalgamated together, which I don't deem very profitable because it will likely break the ability to update stuff. Lastly, the current set-up with lines explaining the notation works rather well and is easy to comprehend, so I don't see much need for such an operation either. &mdash; Lord_Farin (talk) 18:36, 25 February 2013 (UTC)


 * Proposed resolution of the terminology for wosets: Simply put it as a subpage, introduce it as usual and then put a "Remark" section or whatever pointing out that it is an instance of a more general concept. This occurs to me as agreeing with Prime.movers philosophy regarding these matters, see e.g. Definition talk:Inversion Mapping (Topology); and it would appear sensible to admit such duplications on grounds of (ubiquitous) literature presence. &mdash; Lord_Farin (talk) 19:11, 25 February 2013 (UTC)

I never agreed with that conclusion about inversion mapping. I'm not sure I understand your proposal. There's ubiquitous literature using "weak initial segment" to mean "lower closure (of an element)" in the context of well-ordered sets. There seem to be a few sources that extend it beyond that. --Dfeuer (talk) 19:18, 25 February 2013 (UTC)


 * As for why I disagree on that inversion mapping bit: I like to avoid duplication if there isn't a strong reason for it. The only reason for duplicating that definition was to separate the "see alsos" relating to group theory from the ones relating to the theory of topological groups. I think there has to be a better way. We don't define "group" a bunch of different times just because groups are used in algebra, algebraic geometry, algebraic topology, analysis, etc. --Dfeuer (talk) 19:22, 25 February 2013 (UTC)


 * What I mean is that I don't think it's good to enforce "lower closure" everywhere, i.e. I acknowledge the value of admitting "weak initial segment" in the context of wosets (and restraining it to wosets). Also, please consider that there may be things a group of people (such as a wiki community) will never agree on. I was merely quoting Definition talk:Inversion Mapping (Topology) because it was the first example that came to mind (albeit a rather extreme one). &mdash; Lord_Farin (talk) 19:25, 25 February 2013 (UTC)


 * Oh, I had no intention of using it for wosets (except to refer to it on the appropriate definition pages). --Dfeuer (talk) 19:39, 25 February 2013 (UTC)


 * May I ask, in accordance with Definition:Image of Element under Relation is $\bar S_a$ actually $\succeq(a)$? --Jshflynn (talk) 22:49, 25 February 2013 (UTC)


 * It looks that way to me. --Dfeuer (talk) 22:59, 25 February 2013 (UTC)

Progress update
Work is in progress to replace all talk of initial segments (whether they be weak or not) with Definition:Lower Closure and Definition:Strict Lower Closure as appropriate -- except when the ordered set in question is a well-ordered set.

Note that in a well-ordered set, the Definition:Lower Closure and Definition:Strict Lower Closure are by necessity bounded both below and above, whereas when the set is not well-ordered, it may not be bound below, and so therefore may not really be classifiable as a "segment".

As I say, work on this is in progress. --prime mover (talk) 16:31, 17 April 2015 (UTC)


 * I appreciate the direction. As I see it, we've been improving a lot lately. &mdash; Lord_Farin (talk) 18:35, 17 April 2015 (UTC)


 * Further thought: since they are exactly the same thing, I can't see why we need a separate definition for initial segment and weak initial segment so I will replace them both with redirects, with a view towards the note in "also known as". --prime mover (talk) 21:41, 17 April 2015 (UTC)


 * ... except in the end with Definition:Initial Segment I didn't, I kept it as it is, it is too deeply embedded in the mathematical mainstream to countenance a reinvention. --prime mover (talk) 08:36, 18 April 2015 (UTC)