On \(L_{\alpha,\omega}\) complete extensions of complete theories of Boolean algebras (Q1882560)

Let \(T\) be a complete first-order theory of Boolean algebras. It is known that there is a countable ordinal \(\alpha\) such that \(|\{ \text{{Th}}_{L_{\alpha,\omega}}(B): B\models T\text{ and } \| B\|=\aleph_{0}\}|=2^{\aleph_{0}}\), see e.g. \textit{P.~Iverson} [Colloq. Math. 62, No. 2, 181--187 (1991; Zbl 0782.03014)]. The first such limit \(\alpha\) is denoted by \(\alpha(T)\). For every complete first-order theory \(T\) of Boolean algebras which has \(2^{\aleph_{0}}\) nonisomorphic countable models, the author shows that \(\alpha(T)=\omega\cdot 2\) or \(\alpha(T)=\omega \cdot 3\). It answers a question of \textit{S.~Gao} [J. Symb. Log. 66, No. 1, 401--406 (2001; Zbl 1002.03029)]. For the elementary theory of Boolean algebras with an almost principal ideal, invariants (similar to Tarski's invariants) are given, and it is shown that there are exactly \(\aleph_{0}\) complete extensions of this theory. Some open questions are posed, too.