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)


 * Not exactly. I took it out, realized I wasn't sure there wasn't a reason, and put it back. --Dfeuer (talk) 23:21, 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)


 * As I say, give me a moment to think, will you? I'll get back to you in due course, but I'm afraid I have external constraints upon my time at the moment, due to real world interference. --prime mover (talk) 09:37, 13 January 2013 (UTC)


 * Right, here's the lowdown. It appears that the generally-accepted definition of an Definition:Ordered Integral Domain is not just an Definition:Ordered Ring which happens to be an integral domain. Even its wiktionary entry agrees with how we have defined it: as an integral domain with a Definition:Positivity Property. Yes, yes, I know this is not the most general definition that can be conceived, but that is how such an object is conventionally defined.


 * We have two ways of approaching this:
 * a) continue as we are, with the definition that we have posted here of an Definition:Ordered Integral Domain, where the ordering is specifically that induced by the Definition:Positivity Property. This matches mathematical convention throughout the literature generally. Under this regime, an Definition:Ordered Ring which happens also to be an Definition:Integral Domain does not have any special status, and as a concept it has limited appeal.
 * b) be different from convention, and rename Definition:Ordered Integral Domain to Totally Ordered Integral Domain, redefining Definition:Ordered Integral Domain to mean "ordered ring with an ordering compatible with its ring structure". Now, this latter object is rarely defined and explored in the literature, from what I have seen. This is odd, and something for which I can posit the following reasons:


 * 1. The concept of an ordered integral domain being just a "ring with an ordering compatible with its ring structure which is also an integral domain" has not even been thought of, and we are breaking new mathematical ground by establishing it as a definition from which we can open an entirely new field of mathematics hitherto undreamed-of.


 * 2. The object that is an integral domain with a general ordering compatible with its ring structure does not in fact have any useful features which have been determined, despite the 100 or 200 years (? however long) mathematicians have had to explore all its facets. That is, its properties can all be explained in the context of it being an ordered ring and/or an integral domain. Thus no attempt has been made to define it as a "recognised" named object in its own right.


 * My view is that we use approach a) to document this topic, as it matches the status of the field according to existing published sources (or at least, those I have available).


 * If we go down this route, it is of course necessary to add an "also defined as" section to indicate that it is theoretically possible to define an OID as in b) above, but that "for some reason, it's never seen in the literature" (replacing the "never" with "rarely" once such a source emerges).


 * If we go down the route of b), however, we need to make very certain that we are careful to point out that this is not the conventional definition of OID, and that we use TOID for that purpose.


 * I appreciate that the above argument has an observer-bias on it, based on the fact that I am only knowledgeable as far as I have read. While the number of books on mathematics that I have direct access to is actually quite considerable, it is unfortunately a case of "quantity over quality".


 * In order to forestall anyone who says: "Why should my library take precedence over anyone else's preferred source works?" my answer is "I'm sorry, but the reason I rely on my own library is because it's the only works I have access to, and I haven't found anything on line yet which contradicts the sources I've seen. The fact that I haven't actually seen certain other works which other contributors consider "definitive" and "essential" does not in itself necessarily make me a bad person.


 * I freely admit that I can be shot down in flames by the argument "you know nothing about this subject unless you've read these books, and until you have done so you have absolutely no business on even commenting on anything to do with any aspects of mathematics on the internet in any way, shape or form", because I can not think of a counter-argument. Except I will say: Please supply a specific list of those works which you deem essential for my qualification to be considered as part of the on-line mathematical community and I will make the best effort to go and get these books and study them, and in due course I will return here appropriate learnèd. --prime mover (talk) 11:02, 13 January 2013 (UTC)


 * You haven't addressed the matter of "ordered fields", which also are typically defined to be totally ordered.


 * No I haven't. I only addressed ordered integral domains. --prime mover (talk) 14:43, 13 January 2013 (UTC)


 * PW already has established a convention of specifying "totally ordered field" when that is intended.


 * Yes, so it's worth fixing that as well. Give me half a tick, I've had stuff to do today.


 * 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.


 * Why should that be your concern? --prime mover (talk) 14:43, 13 January 2013 (UTC)


 * 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)


 * It may be "only a convenient method of definition", but so what? It's a method of definition which has been proved to be equivalent (IIRC, I haven't checked) to specifying that the ordering is total. That does not invalidate my argument above. --prime mover (talk) 14:43, 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)