More set-theory around the weak Freese–Nation property

Tom 154 / 1997

, Fundamenta Mathematicae 154 (1997), 159-176 DOI: 10.4064/fm-154-2-159-176


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 $ℵ_ω$, 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 $ℵ_1$-Freese-Nation property provided that $μ^{ℵ_0} = μ$ holds for every regular uncountable μ < λ and the very weak square principle holds for each cardinal $ℵ_0 < μ < λ$ of cofinality ω ((Theorem 15). Finally, we prove that there is no $ℵ_2$-Lusin gap if P(ω) has the $ℵ_1$-Freese-Nation property (Theorem 17)

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