On the weak Freese-Nation property of complete Boolean algebras

by Fuchino and Geschke and Shelah and Soukup. [FGShS:712]
Annals Pure and Applied Logic, 2001
The following results are proved: (a) In a Cohen model, there is always a ccc complete Boolean algebras without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a ccc complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of neg 0^#, the weak Freese-Nation property of (P (omega),{subseteq}) is equivalent to the weak Freese-Nation property of any of C (kappa) or R (kappa) for uncountable kappa . (d) Modulo consistency of (aleph_{omega +1}, aleph_omega) ---> (aleph_1, aleph_0), it is consistent with GCH that the assertion in (c) does not hold and also that adding aleph_omega Cohen reals destroys the weak Freese-Nation property of (P (omega),{subseteq}) .


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