Characterize layered analytic ideals via aleph-1 saturation of quotients

Prove that for every analytic ideal I on N, the ideal I is layered if and only if the quotient Boolean algebra P(N)/I is aleph_1-saturated.

Background

The paper introduces layered ideals and shows that all F_sigma ideals are layered and that for every layered ideal I on N, the quotient P(N)/I is aleph_1-saturated. The authors then raise the converse direction as a conjecture for analytic ideals.

Establishing this equivalence would characterize layered ideals intrinsically via the saturation of the associated quotient Boolean algebra, aligning set-theoretic properties of ideals with model-theoretic saturation phenomena studied throughout the paper.