S. Fuchino, L. Soukup:
More set-theory around the weak Freese-Nation property
We introduce a very weak version of the square principle which may hold even
under failure of the generalized continuum hypothesis. Under this weak
square principle, we give a new
characterization of partial orderings with
\kappa-Freese-Nation
property. The characterization is not a
ZFC theorem: assuming Chang's conjecture for aleph_omega, we can find
a counter-example to the
characterization. We then show that, in the model
obtained by adding Cohen reals, a lot of ccc complete
Boolean algebras of cardinality <=\lambda
have the aleph_1-Freese-Nation property provided that
mu^{aleph_0}=mu holds for every regular uncountable
mu<lambda and
the very weak square principle holds for each cardinal
aleph_0<mu<lambda of
cofinality omega. Finally we prove
that there is no aleph_2-Lusin gap if P(omega) has the
aleph_1-Freese Nation property.
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