## 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.