Category talk:Ordered Integral Domains
Why is this a subcategory of Category:Total Orderings? --Dfeuer (talk) 22:41, 12 January 2013 (UTC)
- Because you put it there. --prime mover (talk) 23:19, 12 January 2013 (UTC)
- Oh okay, sorry, I'm with you now.
- So why would this not be a subcategory of Total Orderings? I think it would be suboptimal to stick so strictly to the letter of the category name to restrict it to "total orderings" and not include "totally ordered sets" - the original intention when I envisaged the category was to so include the latter - and so, with that in mind, an ordered integral domain is a totally ordered set. Therefore it belongs as a subcategory of total orderings. --prime mover (talk) 23:32, 12 January 2013 (UTC)
- Are you talking about the definition that appears to require an ordered integral domain to be totally ordered? If so, that doesn't match either ordered ring or, more to the point, ordered field. Since a field is an integral domain, it makes very little sense for an ordered field not to necessarily be an ordered integral domain. --Dfeuer (talk) 23:53, 12 January 2013 (UTC)
- By its very nature, imposing an ordering on an integral domain requires that the ordering be total. You can't have a partial ordering on an integral domain compatible with both operations (in the manner of compatibility as defined for a ring) - this is proved somewhere, I'm not about to find it now. Therefore an ordered integral domain is a totally ordered set, and merits being a subcategory of total orderings. I don't know what you are doing by bringing in ordered fields, except that an ordered field is itself totally ordered for the same reason as is an ordered integral domain. --prime mover (talk) 23:59, 12 January 2013 (UTC)
That's simply not true. As a trivial matter, the diagonal relation on an integral domain is a compatible ordering that is not total. --Dfeuer (talk) 00:10, 13 January 2013 (UTC)
- Okay let me go and think about this, I haven't looked at this area for a while. --prime mover (talk) 00:23, 13 January 2013 (UTC)
It appears common to define an "ordered field" to be what we would call a "totally ordered field", and not to bother defining a "partially ordered field". Googling around gave relatively few references to "partially ordered field", the most promising being this book, which needless to say is not on my shelf. --Dfeuer (talk) 02:06, 13 January 2013 (UTC)
In case you care what I think, I think we should keep our current definition of ordered field and clarify which of our results for ordered integral domains are actually for totally ordered integral domains. --Dfeuer (talk) 02:10, 13 January 2013 (UTC)
- You haven't addressed the matter of "ordered fields", which also are typically defined to be totally ordered.
- PW already has established a convention of specifying "totally ordered field" when that is intended.
- Fixing up PW's much smaller space of proofs about integral domains to match that convention is, I believe, a far smaller task than going the other way around.
- As for the mathematics, the positivity property in this case is only a convenient method of definition, and unless I'm mistaken is just a way of saying that an ordered integral domain is an integral domain which is a totally ordered ring. The reason for that being the requirement that $\forall x\in R:x\in P \lor -x \in P$. --Dfeuer (talk) 14:30, 13 January 2013 (UTC)
There are rather simple correspondences. Note there are other direct correspondences between positivity axioms and order axioms. The axiom that $P \cap -P = \{0\}$ splits more naturally into $P\cap -P \subseteq \{0\}$, which corresponds to antisymmetry, and $0 \in P\cap -P$, which corresponds to reflexivity. --Dfeuer (talk) (some unspecified time)
- Yes we know all that, it's already documented (or ought to be), it's all part of the work establishing the properties of whatever objects we're talking about, whatever they were. What's your point? --prime mover (talk) 18:48, 13 January 2013 (UTC)
Positivity vs. ordering
The point is that saying an ordered integral domain is totally ordered because it's defined from positivity axioms is absurd. The positivity axioms used determine the nature of the relation. There are two reasons to deal with positivity properties at all:
- They may seem more intuitive for some that the notion of a compatible relation.
- Relaxing certain axioms allows for certain generalizations.
Side Note on Ordered fields
There's a perfectly natural definition of an ordered field that lies between ring-ordered field and totally-ordered field: require that if $x$ is positive then so is $x^{-1}$. --Dfeuer (talk) 19:07, 13 January 2013 (UTC)
- Okay okay okay, have it your way. Restructure everything to suit yourself. --prime mover (talk) 20:00, 13 January 2013 (UTC)