### The number of openly generated Boolean algebras

by Geschke and Shelah. [GeSh:558]

J Symbolic Logic, 2008

This article is devoted to two different generalizations
of projective Boolean algebras: openly generated Boolean
algebras and tightly sigma-filtered Boolean algebras.
We show that for every uncountable regular cardinal kappa
there are 2^kappa pairwise non-isomorphic openly generated
Boolean algebras of size kappa > aleph_1 provided there is an
almost free non-free abelian group of size kappa .
The openly generated Boolean algebras constructed here
are almost free.
Moreover, for every infinite regular cardinal kappa we
construct 2^kappa pairwise non-isomorphic Boolean algebras
of size kappa that are tightly sigma-filtered and c.c.c.
These two results contrast nicely with Koppelberg's
theorem hat for every uncountable regular cardinal kappa there
are only 2^{< kappa} isomorphism types of projective
Boolean algebras of size kappa .

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