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

 * 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)


 * Apologies; I have a tendency for excessive parentheses (probably originating from my mind working faster than I can type). --Lord_Farin 18:18, 6 April 2012 (EDT)


 * Sounds good. I note the tedium of the repetitive nature of the task - I get round this by radical copypasta and waiting till my best time of day when there isn't something more interesting / immediate cropping up. --prime mover 19:01, 6 April 2012 (EDT)