  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(\chi). 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_\omega and our very weak square for \aleph_\omega, the
partial ordering [omega_\omega]^{<\omega} (ordered by inclusion) has the
aleph_1-Freese-Nation property.
  On the other hand we show that, under Chang's Conjecture for \aleph_\omega
the 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.
