Source: J. Symbolic Logic Volume 73, Issue 1
(2008), 151-164.
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 2<κ isomorphism types of
projective Boolean algebras of size κ.
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