On Lα,ω complete extensions of complete theories of Boolean algebras

For a complete first order theory of Boolean algebras T which has [inline-graphic not available: see fulltext] nonisomorphic countable models, we determine the first limit ordinal α = α(T) such that [inline-graphic not available: see fulltext] We show that for some [inline-graphic not available: see fulltext] and for all other T‘s, [inline-graphic not available: see fulltext] A nonprincipal ideal I of B is almost principal, if [inline-graphic not available: see fulltext]a is a principal ideal of B} is a maximal ideal of B. We show that the theory of Boolean algebras with an almost principal ideal has [inline-graphic not available: see fulltext] complete extensions and characterize them by invariants similar to the Tarski’s invariants.