On L?,? complete extensions of complete theories of Boolean algebras

For a complete first order theory of Boolean algebras T which has 20 nonisomorphic countable models, we determine the first limit ordinal = (T ) such that |{ThL, (B) : B |= T and ||B||=0}| = 20. We show that for some T s, (T ) = 2, and for all other T s, (T ) = 3. A nonprincipal ideal I of B is almost principal, if {a B : I [harpoonupright]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 0 complete extensions and characterize them by invariants similar to the Tarskis invariants. [PUBLICATION ABSTRACT]