Isomorphy types of superatomic Boolean algebras with one distinguished ideal (Q1972665)

Boolean algebras with one distinguished ideal (\(I\)-algebras) are studied. The author describes isomorphism types of countable superatomic \(I\)-algebras for all elementary types (excluding the type \((\infty, 0, \infty)\)) with finite Frechét rank. Moreover, the author proves that every such \(I\)-algebra is a direct sum of finitely many nonvanishing \(I\)-algebras and that the number of nonvanishing \(I\)-algebras corresponding to a given elementary type is finite.