On the weak Freese-Nation property of complete Boolean algebras

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The following results are proved: (a) 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. (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 □ μ and cof([μ] ℵ 0 ,⊆)=μ + hold for each 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 the consistency of (ℵ ω+1,ℵ ω)↠(ℵ 1,ℵ 0) , it is consistent with GCH that C(ℵ ω) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ℵ ω Cohen reals destroys the weak Freese-Nation property of 〈 P(ω), ⊆ 〉 . These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.