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

For a complete first order theory of Boolean algebras T which has 2 ^א0 non-isomorphic countable models, we determine the first limit ordinal α = α(T) such that |{Th_α,ω(B) : B |= T and ∥B∥ = א₀}| = 2^א0. We show that for some T's, α(T) = ω · 2, and for all other T's, α(T) = ω · 3. A nonracial ideal I of B is almost principal, if {a εB : I ⌈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 א₀ complete extensions and characterize them by invariants similar to the Tarski's invariants.