On the weak Freese-Nation property of complete Boolean algebras

Sakaé Fuchino, Stefan Geschke, Saharon Shelah, Lajos Soukup

The following results are proved:

(a) In a model obtained by adding $\aleph_2$ 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 $\Box_\mu$ and $\mathop{\rm cof}([\mu]^{\aleph_0},{\subseteq})=\mu^+$ hold for each $\mu>\mathop{\rm cf}(\mu)=\omega$, then the weak Freese-Nation property of $\langle{\cal P}\/(\omega),{\subseteq}\rangle $ is equivalent to the weak Freese-Nation property of any of ${\fam\msbmfam\relax C}(\kappa)$ or ${\fam\msbmfam\relax R}(\kappa)$ for uncountable $\kappa$.

(d) Modulo consistency of $(\aleph_{\omega+1},\aleph_\omega)\mathrel{{\rightarrow }\llap{$\rightarrow$}}(\aleph_1,\aleph_0)$, it is consistent with GCH that ${\fam\msbmfam\relax C}(\aleph_\omega)$ does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding $\aleph_\omega$ Cohen reals destroys the weak Freese-Nation property of $\langle{\cal P}\/(\omega),{\subseteq}\rangle $.

These results solve all of the problems listed in Fuchino-Soukup: More set-theory around the weak Freese-Nation property and some other problems posed by S. Geschke.

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