User:Lord Farin/Sandbox

This page exists for me to be able to test out features I am developing. Also, incomplete proofs may appear here.

Feel free to comment.

Over time, stuff may move to User:Lord_Farin/Sandbox/Archive.

Weak/Strict Upper/Lower Closure
These four concepts (cf. Weak Upper Closure) give rise to very similar results.

I suspect it would be most consistent to put them up separately for all four; what do you, reader, think?

Is there a page where it can be expressed that posets admit a duality principle (by passing to the opposite poset, reversing the associated relation)? --Lord_Farin 04:28, 6 April 2012 (EDT)


 * Same thing as we did for upper bound, strict upper bound, lower bound, strict lower bound, etc. Messy and tedious to do, but it looks quite good and works well.


 * There does exist a page somewhere about reversing the ordering, but only in the fairly general terms of inverse relation, and that an inverse ordering is also an ordering. If there's a need for more precision for specific purposes, then feel free to expand it. --prime mover 05:38, 6 April 2012 (EDT)


 * That is Inverse of Ordering is Ordering. --Lord_Farin 10:38, 6 April 2012 (EDT)

Ordering Duality
Let $\Phi$ be a theorem in the language of order theory.

Let $\Phi^{\text{op}}$ be the formula resulting from $\Phi$ by reversing all $\preceq$ signs into $\succeq$ signs.

Then $\Phi^{\text{op}}$ is also a theorem in the language of order theory.

Caution
When higher symbols like $\max$ are used, the $\preceq$ in their definitions also have to be reversed.

Fortunately, this may be accomplished by processing the following changes:


 * $\max \leftrightarrow \min$
 * $\sup \leftrightarrow \inf$
 * Greatest Element $\leftrightarrow$ Smallest Element
 * Maximal Element $\leftrightarrow$ Minimal Element

Pointwise Operations on Mappings
Shortly, I will need a lot of pointwise operations on mappings, so much that I feel there is a case for a page 'Pointwise Operations on Mappings' or something similar. A quick pick (for (extended) real-valued fns):
 * $f+g$, $f g$, $\max(f,g)$, $\sup_i f_i$, $\limsup_i f_i$, $\lim_{i\to\infty}f_i$

and also notions like pointwise limit of mappings $\lim_j f_j = f$

It feels crafted and is unpleasant to keep introducing these every time they are used.

Maybe I could write pages for each of them (specifically for (extended) real-valued functions), and bundle these on 'pointwise operations on real-valued mappings' or something similar. I hesitate a bit because with my natural ability for abstraction, I don't need these myself; they are adequately formulated on Operation Induced on Set of Mappings for anyone capable of the required abstraction. It's just that a (relatively) self-contained treatment (of eg. analysis or measure theory) should not need to delve into the intricacies of abstract algebra when such isn't necessary.

More pages of this sort could be created, and put on ref'd abstract algebra page as examples. --Lord_Farin 07:51, 5 April 2012 (EDT)
 * That is, I would like the reader to comment on this (if he has an opinion). Go ahead, it's free :) --Lord_Farin 16:54, 5 April 2012 (EDT)


 * I would suggest a subpage of Definition:Operation Induced on Set of Mappings which specifically discusses the case where $S$ and $T$ are Definition:Real Functions (or whatever) and none of the baggage of the "set of all mappings" etc., just say $f$ and $g$ are mappings, $f \oplus g (x) = f (x) \oplus g (x)$ where $\oplus$ (or whatever symbol you use) is any operation: "Examples: plus, times, max, sup, etc.". Possibly a different page (or even a different subpage expressing the general result) for the general multifunction $\sup_i f_i$, $\limsup_i f_i$ etc., but I'm not sure how this would be crafted. --prime mover 17:46, 5 April 2012 (EDT)


 * I would not call the page "Definition:Operations Induced on Real-Valued Functions", what I would do is make it a subpage of Definition:Operation Induced on Set of Mappings, e.g. "Definition:Operation Induced on Set of Mappings/Real-Valued Functions"

and have a redirect from Definition:Pointwise Operation.


 * Then as a completely separate page I would put up Definition:Pointwise Limit, in which I would put the entire definition of the pointwise limit as a specific instance of the "pointwise operation".


 * Otherwise there's the danger of trying to cram too much interesting stuff into the same page, which ultimately are just examples of a general rule.


 * Now we're back where we started, because now I can see there's a case for a page called "Pointwise Addition" and "Pointwise Multiplication", which seems to be the multiple-pages-saying-the-same-thing that you were worried about in the first place. But I argue that this approach is not the same as that, because at this stage all you are doing is instancing the general rule as an example. --prime mover 05:31, 6 April 2012 (EDT)


 * Definition:Operation Induced on Set of Mappings/Real-Valued Function is the current try. I have chosen to separate the instantiations in a bright moment of thought-unification with people needing only one kind of pointwise operation. --Lord_Farin 07:39, 6 April 2012 (EDT)


 * Also put up some of the pages with examples. --Lord_Farin 08:38, 6 April 2012 (EDT)


 * It is probably best to put up a specific page for every instance encountered; then for intermediate generalisations (that is, pages fully generalising a certain codomain (like is being done for $\R$ now), and pages fully generalising an operation (eg., addition could be generalised to groups, maybe commutative semigroups, after that it gets awkward to speak of 'addition')), and so on building up to the abstract generality of Definition:Operation Induced on Set of Mappings. Benefit is that most of the proofs can be short, referring to more general ones (giving a nice inherent bound on when a certain pointwise operation is fully generalised: when the proof of a further generalisation wouldn't be identically the same). This gives the potential of, I think, in the order of one hundred pages, if not more. Not sure if I can bring the patience to repeat the same exercise over and over again, but hey, there's no deadline, so what's the problem? Just checking now if I have sensed correctly that this is a desirable direction for PW (where again, I have in mind a reader who's only familiar with a handful of objects pertaining to his field(s) of interest; at least, I want PW to be accessible for people not experts in the terminology and concepts of abstract algebra (as I note I had to become one of the latter to pick up some of the defs on PW)). So, what do you think? --Lord_Farin 17:36, 6 April 2012 (EDT)