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The number of openly generated Boolean algebras

Stefan Geschke·Saharon Shelah
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Abstract

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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Stefan Geschke, Saharon Shelah. 2007-02-21. The number of openly generated Boolean algebras. https://arxiv.org/abs/math/0702600

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