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 (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang's Conjecture for ${\displaystyle {\aleph }_{\omega }}$, we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding Cohen reals, a lot of ccc complete Boolean algebras of cardinality ≤ λ have the ${\displaystyle {\aleph }_1}$-Freese-Nation property provided that ${\displaystyle {\mu }^{{\aleph }_0}=\mu }$ holds for every regular uncountable μ < λ and the very weak square principle holds for each cardinal ${\displaystyle {\aleph }_0<\mu <\lambda }$ of cofinality ω ((Theorem 15). Finally, we prove that there is no ${\displaystyle {\aleph }_2}$-Lusin gap if P(ω) has the ${\displaystyle {\aleph }_1}$-Freese-Nation property (Theorem 17)