Popisfizzy's Site - I am annoyed at partially- and totally-ordered sets.

I am annoyed at partially- and totally-ordered sets. in Off Topic

ID:115068

Jun 26 2011, 7:32 pm (Edited on Jun 26 2011, 9:16 pm)

In order theory, there is a notion of something called a partially-ordered set, or a poset. That is, it is a set S with some operation < such that it is possible to compare some elements a,b, ∈ S : a < b, but it is also possible that some element c ∈ S can not be meaningfully compared to a, b, or both a and b. You could also consider it to be the union (or possibly cartesian product) of a subset S_p with a total order with another set S_q that has a total order, a partial order, or no notion of order.

A total order is a set such that every element can be compared.

As an example, R, the set of all real numbers, has a total order. On the other hand, the complex numbers C can be considered a partial order. You can meaningfully compare the any two real numbers, and any two imaginary numbers, but there is typically no notion of comparing complex numbers in general (not that it can't be applied, it just typically isn't).

The algebra I've been working on can be interpreted as either the cartesian product R×R or R×R×R; it depends on which is most useful. In any case, it is possible to meaningfully compare any two elements in the subalgebras (n,0), (0,n), (-n,-n), as these are isomorphic to the reals (within certain limits). That is, they are the same under the basic arithmetic operations.

The thing is, under the three-dimensional interpretation, any given vector ax̂ is equivalent to -aŷ - aẑ. This means, if a statement like ax̂ > bx̂ is true, it is also true that ax̂ > -bŷ - bẑ, and any basic algebraic manipulation of it like ax̂ + bŷ > -bẑ.

What is driving me insane about this is that the partial order seems to only apply to two vectors of this specific form. I can not figure out a way to meaningfully define ax̂ > bŷ + cẑ, and doing so would necessarily require me to define some sort of new structure relating vectors like (a,b) and (b,a), which seems wonky. Thus, while it really seems like there should be a way to define either a larger partial order or an actual total order, I can't figure it out.

Frustrating. :(

#1 Jun 26 2011, 7:41 pm
Boxcar	I have this problem all the time. Nah jk lol I scraped past algebra with a solid C+, mostly because I cheated on the final exam. To me you are speaking in Latin. Wish I could help though :/

#2 Jun 26 2011, 8:29 pm
Cody123100	Boxcar wrote: I have this problem all the time. Nah jk lol I scraped past algebra with a solid C+, mostly because I cheated on the final exam. To me you are speaking in Latin. Wish I could help though :/ If you're pursuing a CS degree you will encounter set theory soon enough. And you will probably hate it. I know I did.

#3 Jun 26 2011, 9:00 pm
Boxcar	Yeah math isnt exactly difficult for me I just cant stand doing it and dont practice. Even though I understand it fully when its taught my lack of practice always leaves me bewildered on test day because I cant remember how to do it lol.

#4 Jun 26 2011, 9:11 pm
Albro1	Where is the "like" button? That is exactly like me, haha.

#5 Jun 27 2011, 3:56 am
Stephen001	I loved me some set theory, but I suck at more "conventional" calculus.

#6 Jun 27 2011, 11:24 am
Vermolius	Considered asking someone in the field?

#7 Jun 27 2011, 11:38 am
Popisfizzy	Vermolius wrote: Considered asking someone in the field? I have asked them about different things several times. The thing is, I really have a deep down feeling that either the order I want doesn't exist, or it isn't algebraically meaningful, despite this nagging feeling that there is one, and their opinion is pretty much the same.