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 , 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 -Freese-Nation property provided that holds for every regular uncountable μ < λ and the very weak square principle holds for each cardinal of cofinality ω ((Theorem 15). Finally, we prove that there is no -Lusin gap if P(ω) has the -Freese-Nation property (Theorem 17)
@article{bwmeta1.element.bwnjournal-article-fmv154i2p159bwm,
author = {Saka\'e Fuchino and Lajos Soukup},
title = {More set-theory around the weak Freese--Nation property},
journal = {Fundamenta Mathematicae},
volume = {154},
year = {1997},
pages = {159-176},
zbl = {0882.03046},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv154i2p159bwm}
}
Fuchino, Sakaé; Soukup, Lajos. More set-theory around the weak Freese–Nation property. Fundamenta Mathematicae, Tome 154 (1997) pp. 159-176. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv154i2p159bwm/
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