On the weak Freese-Nation property of complete Boolean algebras

The following results are proved: (a) In a model obtained by adding N-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 existen ce of a c.c.c. complete Boolean algebra without the weak Freese-Nation prop erty is consistent with GCH. (c) If a weak form of square (mu) and cof ([mu](N0), subset of or equal to) = mu (+) hold for each mu > cf (mu) = omega, then the weak Freese-Nation p roperty of (P(omega), subset of or equal to) is equivalent to the weak Free se-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo the consistency of (N omega +1, N omega) --> (N-1,N-0), it is co nsistent with GCH that C(N-omega) does not have the weak Freese-Nation prop erty and hence the assertion in (c) does not hold, and also that adding N-o mega Cohen reals destroys the weak Freese-Nation property of C).