This article is devoted to two difierent generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly ae-flltered Boolean algebras. We show that for every uncountable regular cardinalthere are 2 • pairwise non-isomorphic openly generated Boolean algebras of size • > @1 provided there is an almost free non-free abelian group of size •. The openly generated Boolean algebras constructed here are almost free. Moreover, for every inflnite regular cardinalwe construct 2 • pairwise non-isomorphic Boolean algebras of sizethat are tightly ae-flltered and c.c.c. These two results contrast nicely with Koppelberg's theorem in (13) that for every uncountable regular cardinalthere are only 2 <• isomorphism types of projective Boolean algebras of size •.
['Stefan Geschke', 'Saharon Shelah']
The number of openly generated Boolean algebras