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 □_μ and cof([μ]^א0, ⊆) = μ⁺ hold for each μ > cf(μ) = ω, then the weak Freese-Nation property of 〈℘(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) ↠ (א₁,א₀), 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 〈℘(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.