Some Boolean finitely many distinguished ideals II

摘要

We describe the countably saturated models and prime models (up to isomorphism) of the theory Th_(prin) of Boolean algebras with a principal ideal, the theory Th_(max) of Boolean algebras with a maximal ideal, the theory Th_(ac) of atomic Boolean algebras with an ideal such that the supremum of the idea exists, and the theory Th_(sa) of atomless Boolean algebras with an ideal such that the supremum of the ideal exists. We prove than there are infinitely many completions of the theory of Boolean algebras with a distinguished ideal that do not have a countably saturated model. Also, we give a sufficient condition for a model of the theory T_X of Boolean algebras with distinguished ideals to be elementarily equivalent to a countably saturated model of T_X.