Theories of Boolean algebras with distinguished ideals which have no prime model (Q1914758)

The reviewed paper continues the preceding one [reviewed above]. The main results are the following ones: 1) A countable Boolean algebra \(A\) is nonsuperatomic iff at least one of the following conditions is satisfied: (a) there exists an ideal \(I\) such that \(\text{Th} (A;I)\) has no prime model; (b) there exist continuum many ideals \(I\) whose theories \(\text{Th} (A;I)\) are distinct and have no prime model; (c) there exists an ideal \(I\) such that \(\text{Th} (A;I)\) has a prime model but no countably saturated models; (d) there exist continuum many ideals whose theories are distinct and have prime models but no countably saturated models; (e) there exist continuum many ideals whose theories are distinct and have a countably saturated model. 2) If a Boolean algebra \(A\) is superatomic then \(\text{Th} (A;I)\) has a countably saturated model for any ideal \(I\).