Question

Given two infinite sets $X$ and $I$, and a filter ${\cal F}$ on $I$, one defines as usual the equivalence relation $\approx_{\cal F}$ on $X^I$ and obtains the reduced power $Y = X^I / \approx_{\cal F}$.

Question 1 : to what extent do such reduced powers differ when one filter on $I$ is changed to another filter on $I$ ?

Question 2 : consider question 1 in the case of different ultrafilters on $I$, thus in the case of ultrapowers.

Answer 1

Easy differences arise if one allows principal ultrafilters, since the ultrapower of $X$ by a principal filter is canonically isomorphic to $X$, but other ultrapowers are not. Another easy difference arises when $I$ is uncountable, since one filter might concentrate on a countable subset of $I$ and others might not, and this can dramatically affect the size of the reduced power, making them different.

So the question is more interesting when one considers only non-principal filters and also only uniform filters, meaning that every small subset of $I$ is measure $0$.

In this case, under the Generalized Continuum Hypothesis, the ultrapower of any first order structure is saturated, and thus any two of them will be canonically isomorphic by a back-and-forth argument. Without the GCH, it is consistent with ZFC to have ultrafilters on the same set leading to nonisomorphic ultrapowers.

Also relevant is the Keisler-Shelah theorem, which asserts that two first order structures---such as two graphs, groups or rings---are elementarily equivalent (have all the same first order truths) if and only if they have an isomorphic ultrapowers.

Answer 2

Given that you are interested in ultrapowers of $\mathbb{R}$, you might like the following which appears in a joint paper with Kramer, Shelah and Tent.

Theorem: Up to isomorphism, the number of ultrapowers $\prod_{\mathcal{U}} \mathbb{R}$, where $\mathcal{U}$ is a nonprincipal ultrafilter over $\mathbb{N}$, is 1 if $CH$ holds and $2^{2^{\aleph_{0}}}$ if $CH$ fails.

Here $CH$ is the Continuuum Hypothesis. (In the case when $CH$ fails, the relevant ultrapowers are already non-isomorphic merely as linearly ordered sets.) The relevant reference is:

L. Kramer, S. Shelah, K. Tent and S. Thomas Asymptotic cones of finitely presented groups, Advances in Mathematics 193 (2005), 142-173.

Answer 3

Since the question was about reduced powers, not just ultrapowers, it seems worthwhile to point out that, when $F$ is a filter on $I$ but not an ultrafilter, then the reduced power of a field $k$ with respect to $F$ will not be a field, not even an integral domain. Proof: Since $F$ isn't an ultrafilter, $I$ can be partitioned into two pieces $A$ and $B$, neither of which is in $F$. Then the characteristic functions of these pieces represent nonzero elements of the reduced power $k^I/F$ whose product is zero.

Thus, reduced powers of fields modulo non-ultra-filters differ very strongly from ultrapowers, since the latter are fields.

A similar argument shows, for example, that if $X$ is a linearly ordered set with at least two elements, then the reduced power $X^I/F$ is linearly ordered if and only if $F$ is an ultrafilter.

Answer 4

The (latest version of the) question is probably more general than it should be. Since $X$ is allowed to change and since the filters could be principal ultrafilters, the answer is that absolutely anything can happen. Use a principal ultrafilter $F$ on any set $I$, and vary $X$ at will. If the intention was to prohibit principal ultrafilters, then I can't say that absolutely anything can happen, but quite a lot can. For example, given any ultrapower of any $X$, one could change $X$ to a set bigger than that ultrapower; any ultrapower (or any reduced power) of the new $X$ would be bigger than the ultrapower you started with. So it's not clear that anything useful can be said at (or even near) the level of generality of the question.

Answer 5

Let me specify my question. Given two ultrapowers $X^I/F$ and $Y^J/G$ where $X,I,Y,J$ are arbitrary infinite sets, while $F,G$ are ultrafilters on $I$ and $J$< respectively, the questions is to what extent :

1) the cardinals of those two ultrapowers can differ ?

2) those two ultrapowers are not isomorphic when $X$ and $Y$ are fields ?

Answer 6

A more precise, and at the same time, more general form of the question is as follows : Given a reduced power X^I / F, where X and I are arbitrary infinite sets, while F is an arbitrary filter on I, to what extent does the reduced power X^I / F change, when X, I or F are changed ?