On the weak Freese-Nation property of complete Boolean algebras

Annals of Pure and Applied Logic 110 (1-3):89-105 (2001)
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Abstract

The following results are proved: In a model obtained by adding ℵ 2 Cohen reals , there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. Modulo the consistency strength of a supercompact cardinal , the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. If a weak form of □ μ and cof =μ + hold for each μ >cf= ω , then the weak Freese-Nation property of 〈 P ,⊆〉 is equivalent to the weak Freese-Nation property of any of C or R for uncountable κ . Modulo the consistency of ↠ , it is consistent with GCH that C does not have the weak Freese-Nation property and hence the assertion in does not hold , and also that adding ℵ ω Cohen reals destroys the weak Freese-Nation property of 〈 P , ⊆ 〉 . These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 159–176, and some other problems posed by Geschke

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Citations of this work

On the weak Freese–Nation property of ?(ω).Sakaé Fuchino, Stefan Geschke & Lajos Soukupe - 2001 - Archive for Mathematical Logic 40 (6):425-435.
The number of openly generated Boolean algebras.Stefan Geschke & Saharon Shelah - 2008 - Journal of Symbolic Logic 73 (1):151-164.

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References found in this work

A very weak square principle.Matthew Foreman & Menachem Magidor - 1997 - Journal of Symbolic Logic 62 (1):175-196.
On the weak Freese–Nation property of ?(ω).Sakaé Fuchino, Stefan Geschke & Lajos Soukupe - 2001 - Archive for Mathematical Logic 40 (6):425-435.

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