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Some remarks on openly generated Boolean algebras

Published online by Cambridge University Press:  12 March 2014

Sakaé Fuchino
Sakaé Fuchino*
Affiliation:
Institut für Mathematik II, Freie Universität Berlin, W-1000 Berlin 33, Federal Republic of Germany E-mail: fuchino@math.fu-berlin.de
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Abstract

A Boolean algebra B is said to be openly generated if {A : A ≤rcB, ∣A∣ = ℵ0} includes a club subset of . We show:

(V = L). For any cardinal κ there exists an L∞κ-free Boolean algebra which is not openly generated (Proposition 4.1).

(MA+(σ-closed)). Every -free Boolean algebra is openly generated (Theorem 4.2).

The last assertion follows from a characterization of openly generated Boolean algebras under MA+(σ-closed) (Theorem 3.1). Using this characterization we also prove the independence of problem 7 in Ščepin [15] (Proposition 4.3 and Theorem 4.4).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 1 , March 1994 , pp. 302 - 310
Copyright
Copyright © Association for Symbolic Logic 1994

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References

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