User talk:Lord Farin

Orderings on products
I'd like to merge the current Definition:Ordered Product and Definition:Lexicographic Order, extend them to well-ordered index sets, and rename them Definition:Lexicographic Ordering. What's the right way to do such while preserving history, avoiding confusion, etc.? I think Definition:Ordered Product should probably become something of a disambiguation page, pointing to Product Order and Lexicographic Order. --Dfeuer (talk) 21:58, 18 December 2012 (UTC)


 * These will be edits that hit a substantial amount of PW. Therefore, it's probably best to first set up the stuff in e.g. your sandbox area. That way, we can tweak and adjust all we want without affecting the main wiki with immature or incomplete material. I've done this in the past, can't remember what section of the site it was, but it worked quite well. --Lord_Farin (talk) 22:13, 18 December 2012 (UTC)


 * Found something thornier: the site already has a somewhat different notion of lexicographic ordering, which is not on products at all but essentially on strings whose letters are drawn from a single totally ordered set. So I guess we need two different kinds of lexicographic orderings? The kind on strings appears to be isomorphic to a special case of the one on products, since all that's needed is a product using the naturals as the index set, where the totally ordered underlying set is augmented with a new least element, or so I figure. Dfeuer (talk) 01:09, 19 December 2012 (UTC)


 * The one on strings indeed appears to be a special case (similar to $\R^n$ being a special case of Cartesian product). The two are conceptually sufficiently distinct that I'd advise for them to be separate sections (a set-up like Definition:Continuous Mapping, where the real and topological versions are both mentioned). --Lord_Farin (talk) 08:52, 19 December 2012 (UTC)

Can you take a look at User:Dfeuer/Definition:Lexicographic Ordering on Product and the pages it links to? I think they're a decent start on the lexicographic side of things. --Dfeuer (talk) 07:47, 21 December 2012 (UTC)


 * Definition appears to be correct information-wise, and is nice, very general :). However, it's not up to house style (yet). Do you want me to fix that? --Lord_Farin (talk) 15:27, 21 December 2012 (UTC)


 * You're more than welcome to. Small annoyance: there's a page stating (and beginning to prove) the theorem that the lexicographic ordering of the set of all finite sequences on a well-ordered set with at least two elements is not a well-ordering. A similar result holds for lexicographic orderings on products of infinitely many well-ordered sets, each containing at least two elements, and the proof is essentially identical.
 * However, because encoding the finite sequence set as a subset of the product expands each well-ordered set to
 * at least three elements, neither theorem seems to imply the other. Can you think of any nice way to hit both at once, or do we need to keep them separate? --Dfeuer (talk) 16:18, 21 December 2012 (UTC)


 * I think they'd best remain separate. It would be a good idea (and probably not too hard) to try and give a proof of the finite-sequence version with the more general version. --Lord_Farin (talk) 17:37, 21 December 2012 (UTC)


 * I'm not convinced the product version really is strictly more general. I suspect that generalizing "finite sequence" to "set of ordinals less than $n$" may offer a generalization in which the product version can be embedded. --Dfeuer (talk) 22:31, 21 December 2012 (UTC)

Deterministic Time Hierarchy Theorem
Hiya Lord Farin Lord_Farin, I’ve finished reworking Deterministic Time Hierarchy Theorem to conform to the house style and everything. Would you like to take a look and tell me how I did? Thanks :) If this is good, then I think I got the hang of it, so I think I can stop pestering you :) — Timwi (talk) 02:41, 19 December 2012 (UTC)


 * As far as house style goes, it appears to be up to standard now. However the proof itself still lacks rigour and links. This may be due to the field in which the result resides not being covered in enough detail yet. I'll try to keep looking through your edits, but what errors/mistakes you still make appear to be more due to overlooking than a structural flaw in your approach. Of note is that the  command is to be used only when there are operators needing subscripts or appropriate sizing (see Help:Editing for more details); fractions can be covered by  . Cheers! --Lord_Farin (talk) 15:27, 21 December 2012 (UTC)

Product of Ring Negatives: citations
I draw your attention to the fact that Product with Ring Negative has been renamed from Negative Product and split into two. The references from : $\S 1$: Exercise $6$ will therefore need to be adjusted - presumably (as with every other presentation of this material I've seen) Product with Ring Negative directly precedes Product of Ring Negatives. All yours. --prime mover (talk) 09:52, 23 December 2012 (UTC)
 * Amended; thanks for the note. Product of Ring Negatives isn't covered. --Lord_Farin (talk) 22:00, 23 December 2012 (UTC)

Help merging
Hey, could you help me out a bit with the merges I requested for Set Contained in Smallest Transitive Set (where I goofed up and created a second page because I didn't find the first one) and Induction on Well-Ordered Set (where there are two unrelated pages for the result for sets and for classes)? Or should that second one be just an extra rename and some linking? --Dfeuer (talk) 00:16, 30 December 2012 (UTC)


 * I'll look at it tomorrow. Bed's calling. --Lord_Farin (talk) 22:36, 30 December 2012 (UTC)


 * I have ideas for the former, hopefully Asalmon agrees; after that I'll merge the pages. The second one indeed comprises two results that are in different realms. Not sure how to approach that at this point. Again, there is a lot of substandard work in the area and I feel it's a bit pre-emptive to start refactoring parts of it when it's up for a thorough passing over altogether. So I'm inclined to let that rest at the moment. --Lord_Farin (talk) 11:39, 31 December 2012 (UTC)

Integer Addition properties: citations
The pages Integer Addition is Commutative and Integer Addition is Associative have been refactored into separate proofs. I note a citation for Halmos and Givant on both of those - you might want to check they are on the appropriate pages. --prime mover (talk) 13:31, 31 December 2012 (UTC)
 * Done; the stuff serves merely as motivation for introducing rings and is deemed known, not proven. --Lord_Farin (talk) 13:35, 31 December 2012 (UTC)

More STUFF
I could use a bit of help with some STUFF.


 * 1) User:Dfeuer/Reflexive Closure of Transitive Relation is Transitive is, I believe, basically better than Ordering is Strict Ordering Union Diagonal Relation, but it could do with a bit more formal justification.
 * 2) User:Dfeuer/Reflexive Closure of Antisymmetric Relation is Antisymmetric is crap. Maybe we should just stick with the way it's written in Ordering is Strict Ordering Union Diagonal Relation, but I'm not in love with that either.
 * 3) User:Dfeuer/sandbox is full of all sorts of lovely things about ordered groups, compatible relations, etc. Feel free to play around a bit. I'd like to bridge the gap between User:Dfeuer/Operating Repeatedly on Transitive Relationship Compatible with Operation and Properties of Ordered Group/OG5, but that would ideally give Properties of Ordered Group/OG5-like results for an idempotent element (not just an identity element) of a compatible transitive relation, and coming up with a name for that theorem is way beyond my abilities. --Dfeuer (talk) 08:32, 7 January 2013 (UTC)


 * I've fixed 1. and 2. and moved them to main; I hope you like the way I've approached 2.. I'll check out your sandbox page later for more stuff to do. --Lord_Farin (talk) 10:29, 7 January 2013 (UTC)


 * Nice, especially on on 2. Much cleaner presentation. --Dfeuer (talk) 14:37, 7 January 2013 (UTC)


 * The proofs I'm aiming to replace get a bit more formal about unions and things. Do you think that stuff is valuable, or is it simple enough not to bother? --Dfeuer (talk) 14:48, 7 January 2013 (UTC)


 * As you probably noticed, I just crafted Definition:Union of Relations. I aim to produce its natural counterpart about intersections in a minute. I thought it valuable to add these because they allow for a more convenient and explicit crossing from relation theory to set theory. --Lord_Farin (talk) 14:49, 7 January 2013 (UTC)

Deletion request....
Since you're awake, could you please delete User:Dfeuer/OG4? I want to move User:Dfeuer/OG3-4 to that spot now that I found (in Ordered Group Equivalences) something else for User:Dfeuer/OG3. --Dfeuer (talk) 08:39, 8 January 2013 (UTC)


 * Done. --Lord_Farin (talk) 09:19, 8 January 2013 (UTC)


 * Thank ye. I'm very nearly finished with ordered groups. --Dfeuer (talk) 09:24, 8 January 2013 (UTC)

Ordered Group stuff
The following sandbox pages could use a once-over by the master, and then I'd like your help figuring out the best way to organize the transcluding pages:

CRG1, CRG2, CRG3, CRG4, CTR5, OG1, OG2, OG3, OG4, OG5, and the pages they link to. --Dfeuer (talk) 03:52, 9 January 2013 (UTC)


 * After writing this, I found a few glaring errors and omissions. I think I fixed them, but more eyes would be good. --Dfeuer (talk) 07:20, 9 January 2013 (UTC)


 * I'll check it out. It'd be appreciated if you took heed to the tidying I do on the pages you write, so that its amount may be reduced as time progresses. I would rather check your edits and find that they are already matching house style. Your anticipated increased effort is appreciated. --Lord_Farin (talk) 08:59, 9 January 2013 (UTC)


 * I think I'm learning, but slowly, slowly. I'll remember "oh, there was some way this had to be", but not remember how, or where I saw it.... --Dfeuer (talk) 09:21, 9 January 2013 (UTC)


 * I've finished my round along mentioned pages and tidied them (except four pages linked to, will get to them later today, I hope). What exactly do you mean with "transcluding pages"? What do you want to make? An amalgamation page perhaps? --Lord_Farin (talk) 10:36, 9 January 2013 (UTC)


 * Properties of Ordered Group is a very rough draft. Note that there's repetition with Ordered Group Equivalences (which is organized differently, and is less complete), so we'll need to do some merging at some point soon. --Dfeuer (talk) 10:39, 9 January 2013 (UTC)

I assume you see the problem with equation numbering (more accurately: duplication of references). It's not so easy to overcome this. But I'll let you sleep first :). --Lord_Farin (talk) 10:48, 9 January 2013 (UTC)


 * I hadn't. Too sleepy. I have no clue how to fix that anyways. --Dfeuer (talk) 10:52, 9 January 2013 (UTC)

Have you had any more thoughts on how to name User:Dfeuer/Operating on Transitive Relationships Compatible with Operation? I'd like to get that and the ordered group stuff out to the main space and then move on to ordered rings. Unrelatedly, I'm stuck on proving distributivity from De Morgan's laws in a uniquely complemented lattice. Can you maybe give me a tiny hint? I tried proving the special case of
 * $\neg a \vee (a \wedge b) = \neg a \vee b$

and I haven't even managed to do that. --Dfeuer (talk) 02:52, 25 January 2013 (UTC)


 * No further ideas, sadly. I (again) have the feeling there is some terminology lacking, but I don't want to invent something without a source to back it up. A quick attempt to prove the distributivity myself didn't check out. Sorry to disappoint. --Lord_Farin (talk) 08:54, 25 January 2013 (UTC)


 * Despite our lax policy (compared with Wikipedia) concerning original research, I would continue my stance of strongly recommending posting stuff only from established sources. If there's no source, then make very sure of your mathematical ground before adding stuff which you have worked out for yourself. --prime mover (talk) 11:00, 25 January 2013 (UTC)


 * I have learned that the hard way. Some of the "original" research (usually just general pedantry and abstract nonsense) I have posted in my early days here turned out to be not entirely correct, to say the least. It is commendable to try and put stuff in sandboxes before littering the "official" realm. --Lord_Farin (talk) 15:43, 25 January 2013 (UTC)


 * prime mover, I welcome you to check any of my proofs that don't show a stub template, any time. If there is an error in the proof we are discussing here, it shouldn't be at all difficult to find—the proof is short and straightforward. My entire development of properties of compatible relations and ordered groups is best described as "plodding". Unlike the Ordered Group Equivalences page, it does not attempt to be an interesting textbook exercise: it just proves all the forms of each equivalence before trudging on to the next. The approach gives a pleasant symmetry to the proofs of similar results, but it mostly aims to be dull, complete, and reliable. --Dfeuer (talk) 17:19, 25 January 2013 (UTC)

Notations change
Sorry for delaying my answer—I fully understand that you could perceive it as ignoring you—I just wanted to make my edits concise. Of course no hard feelings for your blocking action: all in all my fault as, as you noticed, I'm not a part of "main contributing force"; just thought that change of "subgroup product" into "subset product" name won't make any objections… Here I'd like excuse me once more for my hastiness!

As for giving opinions I leaved some in Definition talk:Internal Group Direct Product and had an answer just recently (which I gladly appreciate) but with no rationale for such statement of the definiton… It might be clear to you for you're part of "maintenance team" but it's quite puzzling for me who is outside this group. I'll try to be more careful when making ammendments (not corrections) in the articles and take into heart your pleas but I'd like to ask you for giving more details about the reasons about solutions you percieve as best. Thanks in advance! joel talk 23:54, 9 January 2013 (UTC)

P.S. I hope that you accept changes I've made as for subset/subgroup product and that they are not a nuissance for the editiorial team…


 * For the purpose of deciding notation and nomenclature, it's probably most efficient to call just me an prime.mover the "main contributing force" - notwithstanding the fact that there has been an increase in users which contribute regularly. Maybe I should've stuck with "maintenance team". The edits pertaining to subset product were correct IMO, no worries on that part.
 * Usually I try to justify particularly any negative calls I make (not speaking for prime.mover here, but take into account that he has led PW through the dark years when self-proclaimed gods would come in and change notation because they didn't like it, and that such has carved certain instincts in him). Recently, we've been increasingly proactive in this regard. Note e.g. the advent of Help:FAQ which should relieve some of the burden of answering the same questions over and over again.
 * In this light the Definition:Internal Group Direct Product page should not be taken as representative. Were such ever to occur again, don't hesitate to kindly ask for an explanation if you feel one is appropriate.
 * Finally, on a side note, it's usually a good idea to reply to a comment on the same page (also for discussions in user talk). This ensures comprehensibility and is most likely to get all those interested notified of the reply (via the watchlist updates - users could be watching your, but not my talk and thus miss your reply). --Lord_Farin (talk) 08:27, 10 January 2013 (UTC)

Trivial question
If
 * $a < b$
 * $b \le c$

or


 * $a \le b$
 * $b < c$

then we can conclude that $a < c$.

What is this rule called? It's not, strictly speaking, transitivity, because neither transitivity of $<$ nor transitivity of $\le$ explains it quite directly. --Dfeuer (talk) 21:22, 14 January 2013 (UTC)


 * No idea. It's so intuitive (since applies to any transitive relation) that it ought to have one, though. I'd be happy to call it transitivity since it does apply to all transitive relations $<$. --Lord_Farin (talk) 21:26, 14 January 2013 (UTC)

double redirects
I have gone through and cleaned up all the double redirects except those from your own (and Dfeuer's) pages - this is to alert you that you might want to sort out those ones. --prime mover (talk) 07:49, 17 January 2013 (UTC)


 * Thanks, fixed. Good job. --Lord_Farin (talk) 08:51, 17 January 2013 (UTC)

Template for extensions
The topic of objects that extend other objects has been in my mind of late: compactification, Dedekind completion, order completion, Cauchy completion, etc. In each of these cases, it's sometimes useful to consider the extension to be the embedding, and it's sometimes useful to consider the extension to be the codomain of the embedding. I attempted, clumsily, to express this in Definition:Compactification. I'm wondering if an improved version of that explanation might make for a good template, or, if that would be too difficult to structure, whether we could come up with a good example of how to explain this which could be adapted for use elsewhere. Or, for that matter, if such a thing already exists somewhere. --Dfeuer (talk) 16:36, 24 January 2013 (UTC)
 * I have no real experience in dealing with the embedding as compactification. I am more inclined to see a possibility for innovation in nomenclature here than to create a template. I would also have no idea as to what this template would express, and how to make it malleable to all intended applications. --Lord_Farin (talk) 21:53, 24 January 2013 (UTC)

Random group theory question
Planetmath claims without proof that a totally ordered group with the order topology is a topological group. I'm not really seeing that, though I could of course be missing something. However, is there a term fora group in which every element has a square root? That sort of ordered group would form a topological group for sure. --Dfeuer (talk) 08:14, 28 January 2013 (UTC)


 * It appears to me that any subbasis element of the order topology is sent to another subbasis element under left and right multiplication, and also under inversion. That'd make it a topological group in my book.


 * I don't know a term for such a group. Perhaps the above comment renders your question obsolete. --Lord_Farin (talk) 09:40, 28 January 2013 (UTC)


 * If by a "square root" you mean an element $y$ such that $y \circ y = x$, then this rings a bell somewhere in the back of my mind, but a cursory search has turned up nothing. $C_3$ is such a group, but (unless I misunderstand what I'm talking about) the Klein-4 group is not. Interesting. --prime mover (talk) 09:49, 28 January 2013 (UTC)


 * I thought about $(\Q, +)$ and $(\Q_{>0},\times)$ (also for $\R$ and additive $\C$, as well as vector spaces over the fields among these). --Lord_Farin (talk) 09:52, 28 January 2013 (UTC)


 * The multiplicative rationals don't have square roots. As for your comment, group multiplication does always send subbasic elements to subbasic elements, but I don't think the inverse images of subbasic elements are necessarily open, unless of course you can explain why they would be. The usual proof that they are (in, say, the reals), relies on being able to take half of something (i.e., a "square root" in a group). And a topological group doesn't require multiplication to be open but rather continuous. --Dfeuer (talk) 14:38, 28 January 2013 (UTC)


 * Ah, the multiplicative rationals are almost surely a topological group because they almost have square roots. For any $q\in \Q_{>0}$ such that $q≠1$ there's an $r$ such that $r^2$ is strictly between $1$ and $r$. --Dfeuer (talk) 14:43, 28 January 2013 (UTC)


 * While realising that my earlier arguments didn't work, I think the result can be proved along the following lines (denote $\mu$ for group operation, $\uparrow$ for strict upper closure (lower closure by duality), and assume for convenience the group is abelian):


 * $(x,y) \in \mu^{-1}(\uparrow z)$. Suppose $\exists y': y \succ y'\succ z \circ x^{-1}$. Then it can be shown that $(x,y) \in \uparrow(z\circ y'^{-1})\times \uparrow(y') \subseteq \mu^{-1}(\uparrow z)$. In the other case, we have $\uparrow (z\circ x^{-1}) = \bar\uparrow y$ by total ordering, where $\bar\uparrow$ is weak upper closure. In that case, $(x,y) \in \uparrow(z \circ y^{-1})\times \uparrow (z\circ x^{-1})$ and also, that set is in $\mu^{-1}(\uparrow z)$ (here we use the set equality established). Done.


 * A proofread on that would be appreciated. --Lord_Farin (talk) 15:22, 28 January 2013 (UTC)

Working on it, but the notation is spinning my head a bit. --Dfeuer (talk) 15:33, 28 January 2013 (UTC)


 * $\mu^{-1}(\uparrow z) = \{(x,y) \in G\times G: x \circ y \succ z\}$, $\uparrow z = \{x \in G: x \succ z\}$, $\bar \uparrow z = \{x \in G: x \succeq z\}$. Hope that helps. --Lord_Farin (talk) 15:39, 28 January 2013 (UTC)


 * Oh, I understand each piece of notation. It's holding them all in my head at once that's the problem :P --Dfeuer (talk) 15:48, 28 January 2013 (UTC)

I put it in the sand at User:Dfeuer/Totally Ordered Group with Order Topology is Topological Group to see if we can flesh it out to a full proof. --Dfeuer (talk) 21:29, 28 January 2013 (UTC)

Proof help
LF, can you please help me out with a proof I'm working out in my sandbox, that a continuous real invertible function has a continuous inverse? --GFauxPas (talk) 13:36, 4 February 2013 (UTC)

Congratulations
Congrats on emptying your sandbox! I've never seen the bottom of it before! --Dfeuer (talk) 17:59, 18 February 2013 (UTC)


 * Don't worry - you can always read the important stuff in its archive... :) --Lord_Farin (talk) 18:05, 18 February 2013 (UTC)

More on compact subspaces of linearly ordered spaces
I think User:Dfeuer/Compact Subspace of Linearly Ordered Space/Revproof3 does the trick, proving that a subset of a LOTS that meets the requisite completeness criteria is in fact a LOTS itself, and therefore that Compact Subspace of Linearly Ordered Space/Lemma applies. --Dfeuer (talk) 18:41, 20 February 2013 (UTC)


 * Seems correct on first superficial reading. Nice. --Lord_Farin (talk) 20:22, 20 February 2013 (UTC)


 * I've also sketched out in my sandbox most of a proof that a GO-Space (definition 3) embeds densely in a LOTS, based loosely on the method of Dedekind cuts. Unfortunately, there are many cases in my formulation (8 different cases to prove transitivity!), so the whole thing is less than elegant. Do you have any simplifications to suggest? --Dfeuer (talk) 20:28, 20 February 2013 (UTC)

the new transclusion s/w
There's one small problem: parent pages of pages transcluded under the new s/w don't appear listed as "transcluded" in the "what links here" page, just as ordinary link directs. --prime mover (talk) 23:07, 22 February 2013 (UTC)


 * Might fix that in due time. Some immediate ideas sprout but I perceive it as being of relatively low priority. &mdash; Lord_Farin (talk) 23:09, 22 February 2013 (UTC)

Ping: initial segment
Definition talk:Initial Segment

Equivalence of lower set definitions
Do you think you could magick up a proof of Equivalence of Lower Set Definitions as dual to Equivalence of Upper Set Definitions? I'm still not so facile with the duality stuff, I'm sorry to say. I know the definitions aren't there yet, but I'll get them up soon. --Dfeuer (talk) 02:23, 26 February 2013 (UTC)


 * P.S., also Lower Closure is Closure Operator, dual to Upper Closure is Closure Operator. --Dfeuer (talk) 02:25, 26 February 2013 (UTC)

Massive transitive closure rewrite
I think it generally works better now, but: --Dfeuer (talk) 21:14, 2 March 2013 (UTC)
 * The transclusions are kind of a mess.
 * The proof of Equivalence of Transitive Closure Definitions (Relation Theory)/Union of Compositions is Smallest seems a tad inelegant. I wonder if there's some way to directly derive that from Recursive Construction of Transitive Closure.
 * Subpage names are not reflected in link names.
 * I'm sure I'm missing some issues.


 * Definition:Transitive Closure (Relation Theory) looks good now. General desired form of equivalence proofs is as on Equivalence of Connectedness Definitions. More comments will appear here as I progress. &mdash; Lord_Farin (talk) 22:26, 2 March 2013 (UTC)


 * Header issue at equivalence proof tackled; the extension I wrote precisely serves this purpose. Although it is currently bugged (TOC doesn't work properly, and same will hold for section editing (but the latter is discouraged anyway) - rest assured, it is being worked on). &mdash; Lord_Farin (talk) 22:49, 2 March 2013 (UTC)


 * Pondering: would it be less hairy to prove definition 3 is smallest, and prove properly that def 3 is equivalent to def 4?-- Dfeuer (talk) 22:57, 2 March 2013 (UTC)


 * Maybe I'm able to assess that once the structure of Equivalence of Connectedness Definitions is into place (and that's a genuine comment, not a poke to remind that such still has to be done). &mdash; Lord_Farin (talk) 23:29, 2 March 2013 (UTC)


 * The structure is different. Transitive closure as smallest is something of an axiomatic definition the others attempt to satisfy. --Dfeuer (talk) 23:35, 2 March 2013 (UTC)


 * I think I see what you mean now. Done. I'm reconsidering the numbering, though. The definitions that strike me as the most useful for most purposes are the definition as the smallest transitive superset and the one I awkwardly/imprecisely called "finite chain". Of course, I'm no grand master of relation theory, so I could be wrong about that. --Dfeuer (talk) 00:13, 3 March 2013 (UTC)


 * Numbering is in some sense arbitrary. I no longer consider it very interesting &mdash; let alone worth the hassle of moving pages in two stages just to juggle them around (to which effect possibly the descriptive subpage names may be an aid). Off to bed now, with the one suggestion that each of the section titles gets a number indication of what it proves (e.g. $(1) \iff (4)$). &mdash; Lord_Farin (talk) 00:19, 3 March 2013 (UTC)


 * Oh, and good job :). &mdash; Lord_Farin (talk) 00:19, 3 March 2013 (UTC)


 * Thanks, and good night. --Dfeuer (talk) 00:32, 3 March 2013 (UTC)

Ordering can be Expanded to Compare Additional Pair
I rewrote big chunks of this page, which I created early on and which was horrible and somewhat wrong. Proof 1 is my own design. It is complete, but could probably use a few tweaks. You may well be able to find a way to cut out some of the manipulations and boilerplate. Proof 2 is basically an incomplete expansion of a handwave on Wikipedia's proof of the Szpilrajn extension theorem. --Dfeuer (talk) 19:40, 3 March 2013 (UTC)


 * Some expansion since then. Proof 2 got refactored into a proof for strict orderings and a slightly handwavy/messy proof from that. --Dfeuer (talk) 00:04, 4 March 2013 (UTC)

Topological closures?
You wanted to do something with them? When you do, please see what can be done about the rather awkward names/lack of transclusion/whatever for Closure is Closed/Closure Operator, Intersection of Closed Sets is Closed/Closure Operator, and Set Closure is Smallest Closed Set/Closure Operator. Also, any fresh approaches to any proofs in the newly-created Category:Closure Operators would be appreciated. --Dfeuer (talk) 00:04, 4 March 2013 (UTC)

But I don't know if they're already here, what names they go by, etc.
But I don't know if they're already here, what names they go by, etc.

1
If an algebraic system is closed, then it is also closed under all finite "compositions" of its operations. Simple example: since the reals are closed under addition and multiplication, they're also closed under the $4$-ary operation mapping $(a,b,c,d)$ to $(a+bc)d$.


 * This is interesting and intuitive. I don't recall ever having seen it (it's too obvious to be in many texts, presumably). Suggest searching for it. It's definitely worth having but I'd rather use the literature terminology (if it exists). &mdash; Lord_Farin (talk) 09:00, 9 March 2013 (UTC)

2
Going the other way, if whenever the results of certain operations have a certain property, then their operands do, then this holds also for operations built from them. For instance, a composition of two idempotent inflationary mappings having $x$ as a fixed point means each of them does. Thus if $x$ is a fixed point of $f \circ g \circ h$ it is a fixed point of each. --Dfeuer (talk) 23:27, 8 March 2013 (UTC)


 * I fail to see how you generalise from that instance to your first sentence. Perhaps you could give another example? &mdash; Lord_Farin (talk) 09:00, 9 March 2013 (UTC)


 * These results don't currently exist on PW but there's nothing to stop you adding them. If you can prove them. The understanding is that this particular venture is your own project, mind, you can't just post other people and ask them to finish it off for you. --prime mover (talk) 09:07, 9 March 2013 (UTC)


 * PM:I was looking for help finding the appropriate names (and theorems, if they were already up). LF: It's really the same thing run in reverse. Another way of looking at the same theorem, I now realize. In the strictly positive reals integers, if $ab+c \preceq 5$ then $a,b,c \preceq 5$--Dfeuer (talk) 15:23, 9 March 2013 (UTC)


 * And I said, the results don't exist on PW. As for a name, "generalised closedness theorem" might be adequate. I was referring to a propensity to start something, lose direction, and then need to ask for help. This can distract contributors from their own projects. --prime mover (talk) 15:58, 9 March 2013 (UTC)


 * Now I understand - thanks. I've been contemplating 1, and it seems that what is being asked here is related to logic, namely the generation of terms in formal languages does something like this. It's kind of a generalization to composition to higher arity. If I have some more time I might crawl through some source works to see if they've got a word for it - probably not, in linear texts these kinds of things usually are left implicit or are not named but simply mentioned in the prose. &mdash; Lord_Farin (talk) 15:42, 9 March 2013 (UTC)

Room for Relevant Links
So as to streamline the reading requests that will appear here following my announcement on main talk, please leave links here in a list format. I will try to read whatever appears here; other pages may well be skipped or be addressed later (much later). TIA for keeping the list tidy and reasonably short. &mdash; Lord_Farin (talk) 22:14, 12 March 2013 (UTC)

Section transclusion
The new transclusion software has been tried out on the (embryonic, still hopelessy messy) Definition:Polynomial (Abstract Algebra) page, where a couple of limitations have arisen:

a) Seems that sections transcluded thus do not appear in the page contents at the top of the page. See the existing (as of this time and date) version of Definition:Polynomial (Abstract Algebra) in which "Polynomial over Field" does not appear there. Is this deliberate, and is there a flag which can be included so as to ensure it does?


 * This unfortunate behaviour (which also occurs with the section tag) is something I have been aware of for quite some time now. It has stopped me from enthusiastically implementing the "software" on a lot of places. There has been a bug filed at MediaWiki for quite some time now; it appears to be a quite deep issue which is perceived as low-priority. I'm not working on it presently; until I find a fix (or someone else does, for that matter), it's probably best to limit its deployment (although any fix will likely be back-end, leaving the invocations unchanged).

b) The paradigm: "Title, waffle, Transclusion" as exemplified on the transclusion of the Definition:Polynomial (Abstract Algebra)/Integral Domain page on that page may not be implemented. I've used that technique once or twice as it allows some explanation / justification of a following section while keeping that explanation underneath the title of what you are then going to expound. Is there a way round this limitation?

Don't let this interrupt your thesis - I may have a solution to b) at least, I just need to try it. --prime mover (talk) 07:36, 16 March 2013 (UTC)


 * For (b), it should be the case that specifying no "title" parameter results in no section title above the transcluded part. For the section tag, this can be accomplished by specifying "notitle = true" (naturally, then, the "title" and "header" fields can be omitted. I hope this is clear enough.


 * My thesis is going rather good at the moment; I really consider my moves of the past week as catalysts for its progress. &mdash; Lord_Farin (talk) 08:43, 16 March 2013 (UTC)


 * Mthx - I can work with that then. Good luck. --prime mover (talk) 09:07, 16 March 2013 (UTC)

Terminology clash
Smullyan and Fitting use the term nest to mean a class of sets which is totally ordered by inclusion. No problem. Where things get dicy is that they define a chain to be a nest which is also a set. This clashes with the order-theoretic notion of a chain. How would you suggest this conflict be resolved? --Dfeuer (talk) 20:55, 25 March 2013 (UTC)


 * This restriction on the term "chain" appears to me as idiosyncratic. Suggest to add a note to the "Also defined as" section of chain but keeping def the same. Then where Smullyan/Fitting write "chain" we can write e.g. "chain of subsets" or something like that if it is to be stressed that we consider $\subseteq$. I don't deem it worth the hassle of putting up a separate case. &mdash; Lord_Farin (talk) 22:06, 25 March 2013 (UTC)


 * I imagine an order-theory type would consider any totally ordered class a chain, set or no. I'm not suggesting it should be added as a separate term; I'm more asking for ideas for more concise descriptions of "nest that is set", particularly for titles. --Dfeuer (talk) 22:16, 25 March 2013 (UTC)


 * Both "Nest" and "Chain" seem appropriate, although perhaps not conveying everything. Other than that, I only mention that a proper way of distinguishing sets from classes would be a non-trivial part of a paradigm for set theory. &mdash; Lord_Farin (talk) 22:23, 25 March 2013 (UTC)

Apologies
I apologize for my outbursts. --Dfeuer (talk) 00:45, 29 March 2013 (UTC)


 * Ok. Please get a grip. The phenomenon seems to recur at a nauseating pace. Now before you would start pointing at PM (which in any given situation may or may not be appropriate), please try some introspection first. (And PM, if you're reading this, please try to take the advice to heart as well). &mdash; Lord_Farin (talk) 08:40, 29 March 2013 (UTC)


 * My interactions on the Definition:Number page were measured, although brusque. As has been noted, I'm not happy about existing pages being amended for the sake of it, purely so the contributor can add his mark (a bit like how a dog marks its territory). Ultimately it may or may not matter on an individual page, but IMO it's far more worthwhile to embark on an area of mathematics which is lacking.


 * And in response to the perfectly reasonable observation that I haven't contributed anything of any worth to ProofWiki myself for the last 2 or 3 years (some might say "ever"), I will regretfully have to concur, with the defence that I have been tidying up the infrastructure (although even that is ultimately a pointless task, and from the point of view of the original contributors, completely unwelcome). --prime mover (talk) 09:00, 29 March 2013 (UTC)

Misc
Can you do me a favor and fix up the names of the subpages of Definition:Transitive Closure (Set Theory)? The problem is described in the talk page.


 * Not going to happen, maintenance duties are still on-hold.


 * All right.... hopefully someone will soon. I made a mis-judgement and can't fix it—grating.

What's your take on whether $\operatorname{On}$ should be considered an ordinal or just similar to one? S&amp;F only consider its elements ordinals. Kelley considers it an ordinal, and calls its elements ordinal numbers. I don't know what other writers do.


 * $\operatorname{Ord}$ is simply the class of all ordinals. As such it's not an ordinal to me, because it is not a set. That it behaves like an ordinal is sufficiently accounted for by the possibility of extending the universe (in the presence of a sufficiently powerful large cardinal axiom) or considering inner models.


 * Well, that's the question: does an ordinal have to be a set, or is there exactly one ordinal which is not a set? What do extending the universe or large cardinal axioms have to do with it?


 * If we have an inaccessible cardinal $\kappa$, the Von Neumann hierarchy $V_\kappa$ up to that cardinal is a model of $\mathsf{ZF}$ as well and is called an internal model. OTOH if we're "in" $V_\kappa$ we can "extend" it by going out of our internal model to the "real" $\mathsf{ZF}$-model.

Are finite ordinals more commonly defined as
 * Ordinals that are well-ordered in the dual ordering or
 * Successor ordinals whose elements are all successor ordinals or
 * Something else?

Our current approach (minimal successor set, which is essentially the approach S&amp;F use as well) doesn't produce a notion of natural number without AoI. --Dfeuer (talk) 17:42, 15 April 2013 (UTC)


 * I haven't seen any approach other than using the Axiom of Infinity; AFAIK no set theory development bothers to avoid that one. Your second characterisation fails on $0$, which is not a successor ordinal.


 * I meant $0$ or &hellip;. Natural numbers, inductive proofs, and finite recursion can be done without infinity, replacing quantification over the set of naturals by quantification over the class of naturals (NBG-style) or using schemas (ZF-style). I believe this sort of thing tends to crop up in constructive theories. Aside from that, it's always nice to have a definition that distinguishes something by its own structure, rather than by a property of a set it belongs to. --Dfeuer (talk) 21:55, 15 April 2013 (UTC)


 * Agreed; I was not trying to smash your ambitions :). &mdash; Lord_Farin (talk) 22:30, 15 April 2013 (UTC)


 * I still hope to be back soon. &mdash; Lord_Farin (talk) 20:06, 15 April 2013 (UTC)

Dimension theorem help
My proof of the Dimension Theorem for Vector Spaces has a gap. I believe it can be filled by this lemma, but I haven't been able to figure out how to prove it (assuming it's true). It's essentially a stronger form of Linearly Independent Subset of Finitely Generated Vector Space:

Let $B$ be a basis of a vector space $V$.

Let $L$ be a finite linearly independent subset of $V$.

For each $x \in V$, $x$ can be expressed uniquely as a linear combination of elements of $B$. (Expression of Vector as Linear Combination from Basis is Unique only shows this for finite $B$, but we should be able to fix that, I think).

Let $\Phi(x)$ be the vectors in the expression of $x$ (with non-zero coefficients).

Then: there is a injection $f: L \to B$ such that $f(x) \in \Phi(x)$ for each $x$.

We know from another theorem that for each $F \subseteq L$, there is an injection from $F$ into $\bigcup \Phi(F)$, and I have the feeling that may be enough to lead to some sort of combinatorial proof (with no further need for any information about vector spaces), but I don't know. The situation can get a bit tricksy:

If $\Phi(x) = \{a, b\}$, $\Phi(y) = \{b,c\}$, and $\Phi(z) = \{a,c\}$, for example, then if you set $f(x) = a$ and $f(y) = c$, then you won't be able to choose a value for $f(z)$.

However, you can set $f(x) = a$, $f(y) = b$, and $f(z) = c$ just fine.

Of course, you could have examples where only some of the $\Phi$ values intersect, etc. Any ideas? --Dfeuer (talk) 22:34, 21 May 2013 (UTC)

My conjectural lemma to the conjectural lemma is at User:Dfeuer/Condition for Injective Choice Function.


 * OK, so I've found out that the general version of this lemma is called Hall's Marriage Theorem. Wikipedia has what claims to be a proof of the finite form in the language of graph theory, which is entirely incomprehensible to me at this juncture. But at least I know it's true. --Dfeuer (talk) 01:46, 22 May 2013 (UTC)


 * Seems like you've taken yourself a nice part of the way already. For the finite case (which seems to be what you're dealing with here anyway) you could perhaps take -- for bases $B, B'$ and $\Phi: B' \to B$ -- the poset $\{F \subseteq B': \text{there is an injective choice function for $\Phi\restriction_F$}\}$ by inclusion, which is finite, hence has upper bounds for chains, and then apply Zorn in the standard way to obtain $B'$ as the only maximal element. I don't know if it follows through, though. &mdash; Lord_Farin (talk) 07:58, 22 May 2013 (UTC)


 * Back up. We don't need Zorn by any means. I believe the dimension theorem is a fairly simple consequence of Hall's Marriage Theorem (for infinite sets of finite sets), which I believe is a straightforward application of the Cowen-Engeler Lemma once the finite case is proven. I've found a few proofs of the finite case. I have not yet, however, found one I actually understand properly. They vary from vague descriptions of algorithms to inscrutable (to me) applications of graph-theoretical concepts. --Dfeuer (talk) 09:18, 22 May 2013 (UTC)


 * OK, I think I've gotten most of the proof chain together now: Hall's Marriage Theorem/Finite Set, Hall's Marriage Theorem/General Set, and Dimension Theorem for Vector Spaces. It's kind of ragged, though. Do you see any obvious ways to improve things (aside from adding necessary links)? --Dfeuer (talk) 01:15, 24 May 2013 (UTC)


 * Line of thought seems good. I have reworded a tiny bit in the dimension theorem, otherwise I see no obvious way to improve things (which does not imply such does not exist, of course). &mdash; Lord_Farin (talk) 07:34, 24 May 2013 (UTC)


 * Feel free to reword more if you think of anything. I think the finite form of the marriage theorem is the hardest piece of this to explain properly, aside from the magical proof of Cowen-Engeler. --Dfeuer (talk) 07:38, 24 May 2013 (UTC)

Question
Is there a page for Fermat Surfaces, maybe it's covered under a more general theorem that I'm missing?


 * There is no such page. Feel free to start one up.


 * BTW: Please sign your posts. --prime mover (talk) 06:31, 3 July 2013 (UTC)

Please help me to complete this proof
I cannot figure out how the sixth step evolved to the seventh step in here, and please help me to complete this proof. Many thanks. Kc kennylau Kc kennylau (talk) 13:05, 3 July 2013 (UTC)


 * There is nothing to justify, as far as I'm concerned. &mdash; Lord_Farin (talk) 13:30, 3 July 2013 (UTC)

What to do about Area of Triangle and Area of Triangle in Terms of Side and Altitude and Area of Triangle in Terms of Two Sides and Angle
--Kc kennylau (talk) 03:56, 7 July 2013 (UTC)
 * 1) Merge Area of Triangle in Terms of Two Sides and Angle to Area of Triangle in Terms of Side and Altitude and delete Area of Triangle in Terms of Two Sides and Angle.
 * 2) Remove Area of Triangle in Terms of Side and Altitude
 * 3) Other options


 * The two theorems clearly have merit of existing on their own. My vote would be to remove the corollary and instate links via the "Also see" section.


 * NB. I'll be on vacation until July 22. Please direct any further questions to PM if you would like an answer before that date. &mdash; Lord_Farin (talk) 21:41, 7 July 2013 (UTC)

Absolute convergence stuff
I added a couple thoughts to Talk:Absolutely_Convergent_Generalized_Sum_Converges. We don't currently have definitions here for normed semigroup or normed group, and I don't have appropriate sources, but the concept is trivial: A normed semigroup is a semigroup with a norm satisfying the triangle inequality. A normed group additionally obeys $\lVert g \rVert = 0$ iff $g$ is the identity and $\lVert g^{-1} \rVert = \lVert g \rVert$. A normed semigroup seems to be the most general context for "absolutely convergent infinite series", a normed commutative semigroup seems to be the most general context for "absolutely convergent generalized sum", and a complete normed abelian group seems to be the most general context in which absolute convergence implies convergence. Requiring a Banach space seems like overkill if the underlying field/division ring never even appears! --Dfeuer (talk) 21:00, 1 August 2013 (UTC)


 * While you are right, there's not much added value if the definitions you mention aren't found in the literature. Maybe in five years, when I get back to the functional analysis department (it's going to happen some day, I'm sure. Just be patient :)). &mdash; Lord_Farin (talk) 21:04, 1 August 2013 (UTC)


 * They are found in the literature; it's just not literature on my shelf. --Dfeuer (talk) 21:08, 1 August 2013 (UTC)


 * My usual standing order: do not just enter random stuff that you think you know, particularly definitions and refinements of existing definitions. If the context is not in place it reduces the comprehension value of the existing material. --prime mover (talk) 21:13, 1 August 2013 (UTC)

I just put up a somewhat rough Positive-Term Generalized Sum Converges iff Supremum. This leads immediately to the conclusion that generalized sums are monotone, in a stronger sense than the theorem you put up by that name. Specifically, the generalized sum over any subset of the index set, not just finite ones, is less than or equal to the total sum, and this holds for complete ordered abelian groups in general, not just the reals. --Dfeuer (talk) 00:24, 2 August 2013 (UTC)


 * I just did a bunch of paper-and-pencil work on some of this stuff on the plane ride to Denver, CO (Hurray! Vacation!). It looks like the progression should begin like this:


 * 1. Let $(L,<,\tau)$ be a LOTS. Then an increasing net in $L$ converges to a point iff that point is the supremum of its image. If the order is Dedekind-complete, an increasing net converges iff it is bounded.
 * 2. Let $(G,\cdot,\le)$ be a totally ordered abelian group, considered under the order topology. Let $\{x_i\mid i\in I\}$ be an indexed subset of (weakly) positive elements of $G$. Then the net mapping each finite subset of $I$ to the sum of the $x_i$ values is increasing.
 * 3. In the situation in (2), a convergent subsum is less than or equal to a convergent sum. If the order is complete, then a subsum of a convergent sum of positive values always converges.
 * 4. A sum including both positive and negative values converges iff both the positive and the negative values have convergent sums, which happens iff the sum is absolutely convergent (absolute value for an ordered group—not a norm).
 * 5. If we end up with a first-countable topology and an Archimedean totally ordered abelian group, I believe that's enough to show that a generalized sum can only converge if all but countably many terms are $0$. Archimedean, if I'm remembering correctly, implies that for each $x>0$, we can only have finitely many terms at least as big as $x$. First-countable, I believe, shows that a sequence can get us close enough to $0$, and we can take a countable union of finite sets to capture all the positive terms. I still need to finish working this step out, and there may be a better way to deal with it, but it certainly works out very easily for $\Bbb R$. --Dfeuer (talk) 21:51, 2 August 2013 (UTC)


 * Interesting, and nice work. I would enjoy seeing it fleshed out (but preferably in your own zone until we find some sources dealing with this mostly functional-analytic construct "generalized sum" in the distinctly more abstract situation of a totally ordered abelian group. &mdash; Lord_Farin (talk) 07:23, 3 August 2013 (UTC)

Proof for Eigenspace_for_Normal_Operator_is_Reducing_Subspace
I would like to provide a short proof for it, based on $\operatorname{ker}(A) = \operatorname{ker}(A^*)$. May I? Should I make a separate lemma for it?


 * By all means, do go ahead and replace the stub notice with your proof. Since you're new to this site, please add a call to the Proofread template. This will let us know that the proof is to be revised with regard to house style (and for correctness, of course, but I presume that won't be a problem). If you get back to me upon adding the proof, we can swiftly deal with that part. &mdash; Lord_Farin (talk) 08:16, 2 August 2013 (UTC)

I wrote proofs for which need to be verified (proofread used). &mdash; Loic
 * Eigenspace for Normal Operator is Reducing Subspace
 * Kernel of Linear Transformation is Orthocomplement of Range of Adjoint
 * Kernel of Normal Operator is Kernel of Adjoint (new theorem)


 * I've dealt with the last two; as the first one uses a lemma I'll have to think a bit on how to approach it. I'll get back to it. In the mean time, please take note of the changes I made to your proof (although by all standards, it was already quite good) -- in particular, you may find it interesting to explore the possibilities of the eqn template. &mdash; Lord_Farin (talk) 07:23, 3 August 2013 (UTC)

Thank you for the improvements.

The lemma used in Eigenspace for Normal Operator is Reducing Subspace could also be considered a further corollary of Kernel of Linear Transformation is Orthocomplement of Range of Adjoint.

Furthermore, is the requirement that $A$ be bounded necessary? As long as $A^*$ exists, the formal manipulations the proofs consist of remain valid. I had matrices in mind as I designed the formal proofs, but https://en.wikipedia.org/wiki/Unbounded_operator#Adjoint seems to allow a generalization to unbounded operators. But I am not comfortable with those finesses (in particualar $\langle A x, y \rangle = \langle x, A^* y \rangle$, used in Kernel of Linear Transformation is Orthocomplement of Range of Adjoint, may not hold without additional hypothesis, although $\langle A^* x, y \rangle = \langle x, A y \rangle$ holds by definition). $\operatorname{ker} A^* = \left({\operatorname{ran} A}\right)^\perp$ may be a slightly more general statement (instead of $\operatorname{ker} A = \left({\operatorname{ran} A^*}\right)^\perp$ claimed in Kernel of Linear Transformation is Orthocomplement of Range of Adjoint).


 * I've finally made the time to deal with the third of your proofs. Because the documentation regarding distributing operators over sets (constructs of the form $A M$ with $A$ an operator and $M$ a subspace fall in this category) I have chosen to adapt the argument to be based on elements rather than on the (nonetheless impeccable) initial approach you chose. I hope you don't mind.


 * Regarding the infinite-dimensional case: we've gotten it for free (the inner product relation you mention follows from the other one by conjugate symmetry of inner products) because we didn't use the dimension of the space in any of the arguments (the only thing may be the existence of a basis, but that relates to AC rather than infinite-dimensionality). The other version relating range and kernel you give results from nothing but substituting $A^*$ for $A$ and using $A^{**} = A$.


 * All in all, you've added some nice proofs, in an area that I'm not set up for covering at this point. So feel free to take on some other stubs in the functional analysis department :). &mdash; Lord_Farin (talk) 17:36, 8 August 2013 (UTC)