Two Fσδ Ideals

We find two Fσδ ideals on N neither of which is Fσ whose quotient Boolean algebras are homogeneous but nonisomorphic. This solves a problem of Just and Krawczyk (1984). We consider Boolean algebras of the form P(N)/I, where I is an ideal on N containing the ideal Fin of finite sets. In [3] Just and Krawczyk formulated several conditions on the ideals I,J that guarantee their quotients P(N)/I and P(N)/J to be isomorphic. By identifying sets of integers with their characteristic functions, we equip P(N) with the Cantor-space topology. We can therefore assign topological complexity to the ideals of sets of integers. In particular, we have Fσ, Fσδ , Borel, and so on, ideals on N. Just and Krawczyk have proved that the Continuum Hypothesis implies that (1) all quotients over Fσ ideals are pairwise isomorphic, and (2) the quotient over the ideal of asymptotic density zero sets, Z0 = {A ⊆ N : lim supn→∞ |A∩n|/n = 0}, is isomorphic to the quotient over the ideal of logarithmic density zero sets, Zlog = {A ⊆ N : lim supn→∞( ∑ i∈A∩n 1/i)/( ∑ i<n 1/i) = 0}. They have also introduced a class of EU-ideals that contains both Z0 and Zlog and proved that under CH all quotients over these ideals are homogeneous and pairwise isomorphic. (A Boolean algebra B is homogeneous if it is isomorphic to BA = {B ∈ B : B ≤ A}, for every A ∈ B \ {0B}.) Motivated by this result, Just and Krawczyk posed the following problem. Problem 1 ([3, Problem C]). Is it true that if I,J are Fσδ and not Fσ and both P(N)/I and P(N)/J are homogeneous, then P(N)/I ≈ P(N)/J ? We will prove that this problem has a negative answer. We will also prove that there is an Fσδ ideal whose quotient is not isomorphic to a quotient over any Pideal. (Recall that I is a P-ideal if for every sequence An (n ∈ N) in I there is an A ∈ I such that An \A is finite for all n.) Received by the editors August 27, 2001 and, in revised form, February 8, 2002. 2000 Mathematics Subject Classification. Primary 54D55, 06E99. The first author acknowledges support received from the National Science Foundation (USA) via grant DMS-40313-00-01 and from the PSC-CUNY grant #62785-00-31. The second author was supported by NSF grants DMS-9803676 and DMS-0102254. c ©2003 American Mathematical Society