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More Set-theory around the weak Freese-Nation property

In this paper, we introduce a very weak square principle which is even weaker than the similar principle introduced by Foreman and Magidor. A characterization of this principle is given in term of sequences of elementary submodels of H(χ). This is used in turn to prove a characterization of kappa-Freese-Nation property under the very weak square principle and a weak variant of the Singular Cardinals Hypothesis. A typical application of this characterization shows that under 2^{\aleph_0}<\aleph_ωand our very weak square for \aleph_ω, the partial ordering [omega_ω]^{<ω} (ordered by inclusion) has the aleph_1-Freese-Nation property. On the other hand we show that, under Chang's Conjecture for \aleph_ωthe partial ordering above does not have the aleph_1-Freese-Nation property. Hence we obtain the independence of our characterization of the kappa-Freese-Nation property and also of the very weak square principle from ZFC.