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The number of openly generated Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Stefan Geschke
Department of Mathematics, Boise State University, 1910 University Drive, Boise, Id 83725-1555, USA Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany, E-mail:
Saharon Shelah
Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick. NJ 08854., USA, E-mail:


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 2κ isomorphism types of projective Boolean algebras of size κ.

Research Article
Copyright © Association for Symbolic Logic 2008

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