Talk:Properties of Ordered Ring

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Many of these properties apply to ordered groups. I would suggest making a separate page for those and linking from here to there. Dfeuer (talk) 22:24, 3 January 2013 (UTC)

Sure. I'm quite convinced many of them will apply to monoids as well... The list goes on and on. Someone with enough spare time may pay attention to this, some day. Perhaps. --Lord_Farin (talk) 22:28, 3 January 2013 (UTC)
There is particular significance for these results being specifically applied to an ordered ring - it's part of the basis for one of the axiomatic derivations of the integers, which themselves form an ordered ring (yes I know: ordered integral domain, but let's not get ahead of ourselves).
Well worth extracting the individual statements in this page (it was written very early on in the life of PW, before it had properly taken shape and the "one page per result" rule had been shaken down). Thus the individual results would of course be established in the context of group / monoid, which would in turn then be transferred to the ring context, and thence to ordered fields.
If anyone actually does refactor this in the context described above, my advice would be to keep this particular page and transclude each of the individual results so as to keep the existing citation and its linear flow intact. --prime mover (talk) 22:40, 3 January 2013 (UTC)
If someone gives me a list of page names, I will make the pages and prove the theorems. Otherwise, I can make a bunch of sandbox pages with no names and someone can name them ....--Dfeuer (talk) 01:48, 4 January 2013 (UTC)

== Which are the properties? ==

These are listed oddly:

1 Blah blah. Hence Bloop.

Is principle 1 the blah blah or the bloop? --Dfeuer (talk) 03:50, 4 January 2013 (UTC)

Time to completely rethink this before splitting up

Some of these properties are generally useful theorems/lemmas. Some of them appear to be completely random facts with three-word proofs. I suspect some of them may be used only to prove the others. I think we need to figure out which of these we actually want and which we don't, and what if any important bits we may be missing. --Dfeuer (talk) 05:23, 4 January 2013 (UTC)

We want them all. If you're not happy about where this is going, then please leave this well alone and work on something else instead. --prime mover (talk) 06:19, 4 January 2013 (UTC)
It may indeed be useful to phrase some of them as corollaries of other, more general properties. But instances of a general theorem can be very useful. --Lord_Farin (talk) 08:40, 4 January 2013 (UTC)

Is User:Dfeuer/OR1 the sort of thing y'all want? --Dfeuer (talk) 19:42, 5 January 2013 (UTC)

Except for a few minor points. I think a more elegant proof of the second two equivalences would use that $\preceq \mathop= \prec \mathop\cup \operatorname{Id}$ (as subsets of $S \times S$). --Lord_Farin (talk) 19:45, 5 January 2013 (UTC)
Meh, never mind. That's not going to be shorter or more elegant on second thought. A few minor points that could well be due to my personal preference remain. Nothing too important. --Lord_Farin (talk) 19:47, 5 January 2013 (UTC)
My proof had some horribly stupid errors. Please check again. --Dfeuer (talk) 19:49, 5 January 2013 (UTC)
Amended to match house style. Though admitted by house style, I find two equations separated by just the four characters "and:" on a line look quite ugly. Furthermore, I reordered the statements at the top to match the order of the proof. Finally, I worked around the interpunction issues as described on Help:Editing/House Style#Punctuation niceties. Also slightly increased number of links because I've grown accustomed to it. Standard for that is: One link per paragraph at least. Hope the explanation helps in accustoming to HS. --Lord_Farin (talk) 19:59, 5 January 2013 (UTC)
Oh, forgot to mention that it may be confusing to use "=" in two different ways in one formula; I decided use $\Delta_S$ instead. --Lord_Farin (talk) 20:01, 5 January 2013 (UTC)