Prime and countably saturated models of the theory of Boolean algebras with distinguished ideals (Q1914757)

Elementary theories of Boolean algebras \((A; I_1, \dots)\) with distinguished ideals \(I_1, \dots\) are studied. Such algebras are called \(I\)-algebras. The author finds a wide class \(K\) of \(I\)-algebras such that for every \(B \in K\), \(\text{Th} B\) has a prime model. It is proved that every countable nonsuperatomic Boolean algebra has continuum many \(I\)-enrichments \(C\) with distinct \(\text{Th} C\) that have prime models but not countably saturated models. Sufficient conditions are found for an \(I\)-algebra to be a prime model. In particular it is proved that if a Boolean algebra \(A\) is superatomic, then \((A;I)\) has a prime model for any ideal \(I\). A sufficient condition is found for an \(I\)-algebra to have no countably saturated models.