The number of openly generated Boolean algebras

Journal of Symbolic Logic 73 (1):151-164 (2008)
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Abstract

This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly ϭ-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 > N1 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 ϭ-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 2<κ isomorphism types of projective Boolean algebras of size κ

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