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.