### 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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