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 User:Dfeuer/OG5, but that would ideally give User:Dfeuer/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)