Background/motivation: I'm investigating the construction of models for a first-order modal system (S5) as products of classical models. Since ultraproducts are all classical models and I need non-classical ones as well, I need to look at reduced products where the filter is not an ultrafilter. This leads me to ask about filters in general:


J.L. Bell & A.B. Slomson, in *Models and Ultraproducts* (p. 116), state and prove:



> 
> Lemma 1.17. Let I be a countable set.
>  Then the collections of non-principal,
>  $\omega$-incomplete, uniform, and
>  regular ultrafilters on I all
>  coincide.
> 
> 
> 


Suppose I alter their definitions slightly so the above properties are all defined for filters in general, then modify the lemma to assert that it holds for filters in general. Would that be true? Can anyone supply a reference to a proof or disproof? Thanks.



Which properties of ultrafilters on countable sets hold for filters in general?
If you use literally the definitions in Bell and Slomson, only changing "ultrafilter" to "filter," and if, as in the lemma you cited, you're interested only in flters on a countable set, then I believe non-principal is equivalent to $\omega$-incomplete, while "regular" is strictly stronger and "uniform" is strictly weaker. Unfortunately, I don't have time right now to check this carefully, so I hope someone will object loudly if I've messed it up. 


Now that I have a bit more time, let me add the counterexamples that justify "strictly". Partition $\omega$ into two infinite pieces $A$ and $B$. The principal filter $F\_0$ generated by $A$ is uniform; that establishes the second "strictly" above. For the first, let $U$ be a nonprincipal ultrafilter that contains $B$, and let $F\_1=U\cap F\_0$. Then $F\_1$ is nonprincipal (the intersection of all the sets in it is $A$, which isn't in it), but it is not regular. (A function $f$ as in Bell and Slomson's definition of "regular" on page 114 would have to send each $a\in A$ to a finite set $f(a)$ that contains all elements $j$ of $\omega$, a contradiction.)


