This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly σ-filtered Boolean algebras.
We show that for every uncountable regular cardinal κ there are 2κ pairwise non-isomorphic openly generated Boolean algebras of size κ > ℵ1 provided there is an almost free non-free abelian group of size κ. The openly generated Boolean algebras constructed here are almost free.
Moreover, for every infinite regular cardinal κ we construct 2κ pairwise non-isomorphic Boolean algebras of size κ that are tightly σ-filtered and c.c.c.
These two results contrast nicely with Koppelberg's theorem in [12] that for every uncountable regular cardinal κ there are only 2κ isomorphism types of projective Boolean algebras of size κ.
