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 $(a+bc)d$.

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)