On the weak Freese-Nation property of complete Boolean algebras

Abstract

The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 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)