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)