Talk:Ideal (set theory)

possible merges

I've put suggested merge tags on talk:null set, talk:negligible set, and talk:small set, which I think are kind of all about this same subject. There are potential distinctions -- sometimes "null" is used distinctively to indicate that we're talking about measure, but not always. Null set could maybe be merged into negligible set if it's desired to keep the discussion of the sets separate from the discussion of the ideals, but honestly I'm not sure why that would be desired; the truth is that the ideals are the more important concept (which is why I wrote this article in the first place, as opposed to, say, expanding one of the three articles mentioned above).

Oh, in case anyone asks, the reason for having this article separate from ideal (ring theory) and ideal (order theory) is that those notions are too general. There are fundamental things you do with ideals on sets, such as take products of them and ask about Rudin-Keisler isomorphisms, that don't obviously generalize to the other sorts of ideals. Ideals on sets are studied in several contexts where the other sorts of ideals would not naturally come up.

Which could also be an argument for keeping null set and negligible set separate; I'm not insistent on the merges. That's one reason I was careful to put the merge tags on the talk pages. But it does seem that maybe we have too many articles here on nearly identical topics. Discussion invited. --Trovatore 06:33, 6 August 2007 (UTC)[reply]

I'd like to keep Null set separate because of its special measure meaning and its importance to analysis. The others should probablu be merged. CRGreathouse (t | c) 13:02, 6 August 2007 (UTC)[reply]

I have to say that I don't think ColdFusion650's bold merges really worked. I'm not sure exactly what I was suggesting should happen -- mainly I wanted to get a discussion going about whether there were too many articles and what to do about it -- but it certainly wasn't just dumping large sections of the articles into this one. Unless someone has a better idea I will probably revert those changes later today. --Trovatore 16:58, 8 August 2007 (UTC)[reply]

The information is here. It just needs to be smoothed out so that the integration is more seamless. From my reading of the article, Ideal set, null set, and negligible set are synonymous. Some of this is over my head as Calculus 3 is as far as I've gotten in school so far. ColdFusion650 17:03, 8 August 2007 (UTC)[reply]

Well, but the "smoothing out" is actually the hard work. Right now the merged material appears as a disjointed collection of facts, and does not appear to be about the title topic. I want to keep this article primarily about ideals, not about measure theory -- if the measure theory needs to be maintained with the null set material then maybe it should not be merged. --Trovatore 17:14, 8 August 2007 (UTC)[reply]

Right. The wholesale dumping of text definitely wasn't very helpful. What might be useful would be writing some new content about small/negligible sets in this article and then redirecting those pages here. Or maybe not. —Ilmari Karonen (talk) 01:02, 9 August 2007 (UTC)[reply]

Trivial examples

Nice work! It did occur to me that, to make the article accessible to a broader audience, it might be a good idea to include some even simpler introductory examples. The one that comes immediately to mind would be the trivial ideal on X consisting of all subsets of some set B ⊂ X. Perhaps there are others, too — I'm not really very familiar with the subject.

Also, shouldn't there be a link to Filter (mathematics) somewhere in the article? —Ilmari Karonen (talk) 14:34, 6 August 2007 (UTC)[reply]

I just added the example above to a new "General examples" section, and moved the finite subset ideal there as well since it applies not just to the natural numbers. As I wrote in my edit summary, please feel welcome to improve my changes in any way you see fit (up to and including reverting them, if necessary). —Ilmari Karonen (talk) 20:00, 6 August 2007 (UTC)[reply]

Slightly different definition

Some authors add the condition $X\notin {\mathcal {I}}$ to the definition of an ideal. Other authors use the same definition as in the article and call ideals fulfilling this condition proper ideals. See here for an overview (and feel free to add more references if you think it is needed and you have some at hand). Do you think this should be mentioned in the article? --Kompik (talk) 11:36, 24 July 2009 (UTC)[reply]

Yes, we should mention this in the article, at least so that people know to watch out for the different conventions. — Carl (CBM · talk) 12:57, 24 July 2009 (UTC)[reply]

Hmm, another convention to watch out for, if you're keeping track, is that it is sometimes assumed that all finite sets (equivalently, all singletons) are in the ideal. This is just to avoid trivialities and is usually mentioned when used, so perhaps it's not worth bringing up here. --Trovatore (talk) 00:57, 25 July 2009 (UTC)[reply]

I have seen some papers where ideal containing all singletons is called an admissible ideal. Quick search in google books did not yield some book references for this. By the way the books I have checked usually defined filter and ideal in the dual way; i.e., if the authors included the condition

\emptyset \notin {\mathcal {F}}

in the definition of a filter, they also used

X\notin {\mathcal {I}}

in the definition of an ideal and vice-versa. Perhaps Wikipedia should also be consitent in this? I am not sure whether it would influence other articles too much? (Are there many articles using set-theoretical ideals?) --Kompik (talk) 10:49, 25 July 2009 (UTC)[reply]

I don't think we have a dedicated article on filter (set theory). However, if one talks about ultrafilters on a set, properness is always assumed, of course. It's very hard to be consistent between articles, for several reasons; it's easier to just explain everything in each article. — Carl (CBM · talk) 13:14, 25 July 2009 (UTC)[reply]

In the article filter (mathematics) there is a section filter on a set. (On the talk page, quite a long time ago, a split of the article was suggested.) However, I think that you're right about the problems with keeping so many articles consistent. --Kompik (talk) 14:55, 25 July 2009 (UTC)[reply]

It came to my mind that I do not know where on Wikipedia we discuss the relationship between filters, ideals, and finite-additive two-valued measures, and the relationship between completeness of the filters/ideals and additivity of measures. Is this anywhere? — Carl (CBM · talk) 17:03, 25 July 2009 (UTC)[reply]

This google searches [1] [2] suggest that the only expository article that mentions ultrafilters as two-valued measures is ultrafilter. --Kompik (talk) 06:40, 28 July 2009 (UTC)[reply]