On the weak Freese-Nation property of complete Boolean algebras

The following results are proved: (a) In a model obtained by adding א2 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 algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of 2μ and cof([μ] א0 ,⊆) = μ+ hold for eavch μ > cf(μ) = ω, then the weak Freese-Nation property of (P(ω),⊆) is equivalent to the weak Freese-Nation property of any of C(κ) or R(κ) for uncountable κ. (d) Modulo consistency of (אω+1,אω) → (א1,א0), it is consistent with GCH that C(אω) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding אω Cohen reals destroys the weak Freese-Nation property of (P(ω),⊆). These results solve all of the problems listed in Fuchino-Soukup [5] and some other problems posed by S. Geschke. ∗The first author was partially supported by Grant-in-Aid for Scientific Research (C) No. 10640099 of the Ministry of Education, Science, Sports and Culture, Japan. Some of the results here, in particular earlier versions of the results in Section 5, were included in the second author’s Ph.D. thesis [7]. This paper is [FGShS:712] of the third author’s publications list. His research is supported by “The Israel Science Foundation”. The fourth author was partially supported by Grant-in-Aid for JSPS Fellows No. 98259 of the Ministry of Education, Science, Sports and Culture, Japan, and by Hungarian National