## 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 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 and
hold for each
, then
the weak Freese-Nation property of
is equivalent to the weak Freese-Nation property of any of
or
for uncountable .

(d) Modulo consistency of
, it is
consistent with GCH that
does not have the
weak Freese-Nation property and hence
the assertion in
(c) does not hold, and also that adding Cohen reals destroys
the weak Freese-Nation property of
.

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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