We study reduced products Formula: see text of countable structures in a countable language associated with the Fréchet ideal. We prove that such Formula: see text is Formula: see text-saturated if its theory is stable and not Formula: see text-saturated otherwise (regardless of whether the Continuum Hypothesis holds). This implies that Formula: see text is isomorphic to an ultrapower (associated with an ultrafilter on Formula: see text) if its theory is stable, even if the CH fails. We also improve a result of Farah and Shelah and prove that there is a forcing extension in which such reduced product Formula: see text is isomorphic to an ultrapower if and only if the theory of Formula: see text is stable. All of these conclusions apply for reduced products associated with Formula: see text ideals or more general layered ideals. We also prove that a reduced product associated with the asymptotic density zero ideal Formula: see text, or any other analytic P-ideal that is not Formula: see text, is not even Formula: see text-saturated if its theory is unstable.
Bondt et al. (Fri,) studied this question.