On the weak Freese-Nation property of complete Boolean algebras Original Research Article

The following results are proved: (a) In a model obtained by addingView the MathML sourceCohen 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 □μandView the MathML source hold for each μ>cf(μ)=ω, then the weak Freese-Nation property ofView the MathML sourceis equivalent to the weak Freese-Nation property of any ofView the MathML source or View the MathML source for uncountable κ. (d) Modulo the consistency ofView the MathML source, it is consistent with GCH thatView the MathML sourcedoes not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that addingView the MathML sourceCohen reals destroys the weak Freese-Nation property ofView the MathML source. 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.