Definition talk:Archimedean Property/Ordering

If there's something wrong, please feel free to correct it. I remember nothing about this page. It was probably when I tidied up some kitchen-sink page put together by a new contributor but there's no record now of where the content originated. --prime mover (talk) 19:53, 27 October 2012 (UTC)

Why does this have to be a semigroup by the way? --prime mover (talk) 06:34, 28 October 2012 (UTC)

Correct definition

See the discussion in Definition talk:Archimedean Property/Norm for my reasoning behind why I reverted.

Specifically, I am not sure that your definition of this property is accurate. The specific point about the archimedean property, from how I understand it, is that there is some multiple (power, call it what you will) of any element which will exceed any other given element. Thus the reals under addition are archimedean: $\forall x, y \in \R: \exists n \in \R: n x > y$ but the reals under multiplication are not archimedean.

What are your sources? If they differ specifically from what is defined here, then you are invited to set up a subsection or subpage specifying the differences, or the specific definition when applied to certain types of structure (most noline sources I've found do not generalise beyond real vector spaces when defining norms in the first place) in the way that such pages are structured on this site. --prime mover (talk) 07:42, 28 October 2012 (UTC)

Concerning the norm, I thought it is possible that $n \left({a}\right) = 0$. You said that "for sufficiently high $m$, the norm of the power of any element will exceed the norm of any given element", so what is the condition $n \left({a}\right) < n \left({b}\right)$ for? What I did parallels this Wikipedia page. Where did you get the current definition from?

Concerning the orderings, I thought that it is $\forall x, y \in \R: x > 0 \implies \exists m \in \N_{>0}: m \cdot x > y$. Of course, a general semigroup need not have an identity (such as $0$ in $\left({\R, +}\right)$), so this can't be used as the general definition. We should create a page using this definition (i.e., $\forall x, y \in G: e \prec x \implies \exists m \in \N_{>0}: y \prec x^m$) specifically for totally ordered groups, but I haven't gotten to it yet.

So I came across this page (for totally ordered semigroups). I'm quite sure that it is equivalent to the definition for totally ordered groups, when applied to groups.

By the way, is the notion of Archimedean-ness even considered for non-associative structures? Do you have a source treating the concept in this much generality? --abcxyz (talk) 15:38, 28 October 2012 (UTC)

Like I say, I can't remember where it came from. All I know is it came from somwwhere. What's your take: delete everything unsourced? Fair enough, I have no opinion on the matter. --prime mover (talk) 17:23, 28 October 2012 (UTC)