Definition talk:Strict Upper Closure

Parentheses
Can't we drop the parentheses like we do for other unary operators? ${\uparrow}z$ is just as clear as ${\uparrow}(z)$, no? --Dfeuer (talk) 00:57, 29 January 2013 (UTC)


 * No. --prime mover (talk) 06:20, 29 January 2013 (UTC)


 * Do you realize that "No" is not actually a useful response? —Dfeuer (talk) 06:29, 29 January 2013 (UTC)


 * No --prime mover (talk) 07:02, 29 January 2013 (UTC)


 * It's the answer you should have been expecting: it's a suboptimal strategy to sacrifice clarity of presentation to suit your personal convenience. --prime mover (talk) 07:02, 29 January 2013 (UTC)


 * I guess we have to go around and replace every instance of $-x$ with $-(x)$. Damn, that's going to be annoying. --Dfeuer (talk) 07:10, 29 January 2013 (UTC)


 * In certain circumstances, your suggestion may be of use.


 * There are indeed instances where we are not consistent in our use of brackets. In the case of $-$ this is because of global convention. However, we have established that the notation for mappings on this site is $f(x)$ rather than $fx$, however prevalent in the literature the latter. The operator ${\uparrow}(z)$ is no different. Clarity suffers if ${\uparrow}z$ is allowed to replace ${\uparrow}(z)$. The fact that it irritates you to write brackets is just one of those things you're going to have to suck up. --prime mover (talk) 07:54, 29 January 2013 (UTC)


 * Negation, addition, multiplication, powers, roots, factorials, inverses, floor, ceiling, composition, division, etc., etc., etc., all break this pet rule of yours. We don't write $!(x)$.  We don't write $\circ(x,y)$. We don't write $\lfloor\rfloor (x)$. Blah blah blah. The entire purpose of notation is convenience. --Dfeuer (talk) 08:06, 29 January 2013 (UTC)


 * I would contend that in certain situations the use of brackets can actually clarify the presentation; binding priorities are practised with reason in the logic department. Nonetheless consistency would suffer if the parens were dropped.
 * There could be a case developed in favour of omitting brackets in the spirit of "eliminating visual junk from the page" as is done with punctuation in displayed equations. I am currently not so inclined but as and when we run out of immediate huge tasks, feel free to bring up the discussion again. Thus, "not now" is the answer. --Lord_Farin (talk) 08:11, 29 January 2013 (UTC)


 * I am getting extremely frustrated with the fact that it seems that editing math on PW always involves a tremendous amount of typing \left({}\right) and then trying in vain to read the result of doing so when there's any amount of nesting. I would in any case still prefer ${\uparrow}x$ to ${\uparrow}(x)$, but the left-right rule makes me feel ever so much more strongly --Dfeuer (talk) 08:19, 29 January 2013 (UTC)


 * I am sorry you feel that way, but I suggest you may wish to try out some readily-available editing help tools. Some of these may allow you to set up macros which you may e.g. type ctrl-L and get \left({ and crtl-R to get }\right). It's a suggestion which I haven't tried myself because I tend to use cut-and-paste, being rather more old-school. --prime mover (talk) 08:31, 29 January 2013 (UTC)

Did you read the part about how I can't READ what I've written, and therefore have an awfully hard time figuring out what the heck is going on when things don't match up properly? As much as I hate the extra typing (and yes, I do), I hate having to read the resulting mess even more. --Dfeuer (talk) 08:34, 29 January 2013 (UTC)


 * Such is life. --prime mover (talk) 08:41, 29 January 2013 (UTC)


 * You may profit from an external editor (as PM suggests) with a custom highlight scheme. With the Firefox "It's all text" plugin (which I use sometimes) you can set an external editor yourself, for example your favourite TeX editor, which will undoubtedly highlight matching braces and left-right pairs. This will make editing and understanding code easier. For input, there are enough tools binding macros to key combinations (e.g. AutoHotKey). Takes a few moments to set this all up but if you are really frustrated it may help you. --Lord_Farin (talk) 09:21, 29 January 2013 (UTC)

Naming and notation
Does anyone other than PW use this notation? An Internet search turned up exactly one non-PW hit for "strict lower closure", a paper using a double-downarrow symbol for it. A search for "strict upper closure" turned up zero non-PW sources. However, it does seem common to use $\uparrow$ and $\downarrow$ for what we call "weak upper closure" and "weak lower closure" (and others just call "upper closure" and "lower closure") as well as their extensions to sets. The names we're using don't even make sense&mdash;the strict versions aren't closures of any kind. --Dfeuer (talk) 07:57, 20 February 2013 (UTC)


 * You're basically saying (surprisingly more diplomatically than usual) "The notation and name are complete rubbish" without offering an alternative. Please, when suggesting there is room for improvement, consider making a further suggestion as to which direction you think the improvement lies. Alternatively, finish off some of the many jobs that you've started but have not yet finished. --prime mover (talk) 08:59, 20 February 2013 (UTC)


 * I wasn't yet offering an alternative because I did not know the usual terminology or notation yet. It appears that the terms "up-set", "down-set", "upper closure" and "lower closure", and the symbols $\uparrow$ and $\downarrow$ are commonly used for what we have been calling weak upper and lower closures. It appears that the terms "strict up-set" and "strict down-set" and the symbols $\dot\uparrow$ and $\dot\downarrow$ are used for what we've been calling "strict upper closure" and "strict lower closure". The symbol $\updownarrow$ is used for convex closure. There are obviously too many pages using these operators to switch them all to the more common forms in one go. My suggestion, then, would be to introduce $\dot\uparrow$ and $\dot\downarrow$, switch pages over to those as we are able, and then once that is complete, redefine $\uparrow$ and $\downarrow$ to their common meanings and start replacing the barred forms. --Dfeuer (talk) 22:16, 20 February 2013 (UTC)


 * As long as you leave the originally defined name and notation, when you change them all over, as an "also known as" and "also denoted as", you'll be all right. --prime mover (talk) 22:21, 20 February 2013 (UTC)


 * Would you be so kind as to disclose the sources of this usual terminology and notation, particularly the latter? Not to criticize (I value your search) but I'd rather have some reference before indulging in this piece of work. The terms "up-set" and "down-set" aren't very appealing to me; certainly less so than upper and lower closure. Literature wins, though. --Lord_Farin (talk) 22:22, 20 February 2013 (UTC)


 * If you just Google "strict up-set" or "strict up set" you'll get a bunch of results using this or similar notation (plain arrow for upper closure; marked arrow for strict up-set). If you google "strict upper closure" and "strict lower closure" you get virtually nothing but us. Furthermore, "strict upper/lower" closure is not a closure operator (it's not idempotent!) so that terminology doesn't make much sense. --Dfeuer (talk) 22:26, 20 February 2013 (UTC)


 * I see. Apparently $\uparrow\hspace{-11mu}\text-$ is also used but it's awkward to produce &mdash; I expect such to get (much) worse when trying to use it in more complicated constructs. --Lord_Farin (talk) 22:42, 20 February 2013 (UTC)

Come to think of it, what's wrong with Definition:Initial Segment and (presumably) Final Segment? In Devlin it's actually called a "segment" as his thesis is ultimately concerned only with wosets and therefore only the lower set. I can't find my Halmos atm but am fairly sure that's where I got "initial segment" from.

As we're working towards only using terms backed up by literature and sources, my suggestion trumps everyone else's as I'm the only one to provide a hard-copy source beyond a vague "google for it". --prime mover (talk) 07:05, 21 February 2013 (UTC)


 * ... thus the notation for $\uparrow \left({a}\right)$ and $\downarrow \left({a}\right)$ become the typographically straightforward $S^a$ and $S_a$. Then all our troubles will vanish in a puff of smoke. --prime mover (talk) 07:08, 21 February 2013 (UTC)


 * Because that terminology and notation is usually used only for wosets, while the arrow notation is used for other ordered sets. As for published works, some of those (shock!) turned up in the Google search. You should try it. --Dfeuer (talk) 07:12, 21 February 2013 (UTC)


 * Since you're the one proposing these notations, the responsibility for providing the source works backing them up is yours. It's on a par with: when writing a dissertation, you don't put a note in the references section saying "Google for it". --prime mover (talk) 07:31, 21 February 2013 (UTC)


 * Will do. --Dfeuer (talk) 07:35, 21 February 2013 (UTC)


 * The word "segment" also has geometrical connotations suggesting total ordering. --Dfeuer (talk) 07:22, 21 February 2013 (UTC)


 * The notation $S_a$ is compromised because we use $S$ to denote sets. Also, $S_{\sup \text{uglyexpression}}$ is not really improving readability... and because both the super- and the subscript version would be used there would be no possibility to resort to something in-line. --Lord_Farin (talk) 09:05, 21 February 2013 (UTC)