In Gillman and Jerrison's book, "Rings of Continuous Functions", they show a nice relationship between the set of z-ideals in C(X) and the set of filters on X. One can go for with this relationship; for example, proving the smallest measurable cardinal (if any exist) is strongly inaccessible, seems to be better understood from a filter perspective (rather than ideal).

Now instead of considering the spectrum of a ring, is there any usefulness to consider the set of filters (which consist of algebraic sets), and defining a topology on it? It just seems like some things (like analytic stuff) would be more natural working in a filter setting.

I suppose for a variety V, you could work with something like Filt(V) instead of Spec(V). Where Filt(V) consists of fixed filters of a certain type (each filter would correspond to a variety). Is there any study of this kind? Or is this analogy not very useful?

  • $\begingroup$ Notice, that for a tychonoff space X, the stone-cech compactification of X characterizes the set of ultrafilters on X (i.e. the points of BX are in one-one correspondence with the ultrafilters on X). This seems like it could be a useful generalization of the spectrum of a ring (at least for the maximal ideals of it). $\endgroup$ – six Oct 22 '12 at 5:25
  • $\begingroup$ filters and ultrafilters are used in real algebraic geometry, which has more connections to logic than the classical one, it seems. $\endgroup$ – Dima Pasechnik Oct 22 '12 at 5:26
  • $\begingroup$ It seems like I remember seeing something about how the Stone-Cech Compactification characterizes the spectra of a C*-algebra (which generalizes spectrum of ring). However, this might only be useful (and true) when X satisfies some specific conditions (completely regular). $\endgroup$ – six Oct 22 '12 at 5:42
  • $\begingroup$ Posted on math.SE: math.stackexchange.com/questions/218242 $\endgroup$ – Martin Brandenburg Oct 22 '12 at 8:16
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    $\begingroup$ six: (1) In the first paragraph of the question, "nonmeasurable" should be "measurable". (2)In your first comment, the canonical bijection between ultrafilters on $X$ and points in its Stone-Cech compactification is only for discrete $X$. In general, the Stone-CEch compactification is a quotient of the space of ultrafilters (and depends on the topology of $X$). (3) I don't see much difference between working with a filter $F$ and working with the ideal of complements of elements of $F$ (except that you might have to move to a different lattice, e.g., open sets vs. closed sets). $\endgroup$ – Andreas Blass Oct 22 '12 at 11:40

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