### 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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