### The number of openly generated Boolean algebras

Stefan Geschke and Saharon Shelah
Source: J. Symbolic Logic Volume 73, Issue 1 (2008), 151-164.

#### Abstract

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 isomorphism types of projective Boolean algebras of size κ.

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Primary Subjects: 06E05
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1208358746
Mathematical Reviews number (MathSciNet): MR2387936
Digital Object Identifier: doi:10.2178/jsl/1208358746
Zentralblatt MATH identifier: 1137.06006

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