Saturation of reduced products

In this paper, 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.