# More set-theory around the weak Freese–Nation property

Fundamenta Mathematicae (1997)

- Volume: 154, Issue: 2, page 159-176
- ISSN: 0016-2736

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topFuchino, Sakaé, and Soukup, Lajos. "More set-theory around the weak Freese–Nation property." Fundamenta Mathematicae 154.2 (1997): 159-176. <http://eudml.org/doc/212231>.

@article{Fuchino1997,

abstract = {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)},

author = {Fuchino, Sakaé, Soukup, Lajos},

journal = {Fundamenta Mathematicae},

keywords = {Lusin gap; weak version of the square principle; partial orderings with -Freese-Nation property; Cohen reals; ccc complete Boolean algebras},

language = {eng},

number = {2},

pages = {159-176},

title = {More set-theory around the weak Freese–Nation property},

url = {http://eudml.org/doc/212231},

volume = {154},

year = {1997},

}

TY - JOUR

AU - Fuchino, Sakaé

AU - Soukup, Lajos

TI - More set-theory around the weak Freese–Nation property

JO - Fundamenta Mathematicae

PY - 1997

VL - 154

IS - 2

SP - 159

EP - 176

AB - 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)

LA - eng

KW - Lusin gap; weak version of the square principle; partial orderings with -Freese-Nation property; Cohen reals; ccc complete Boolean algebras

UR - http://eudml.org/doc/212231

ER -

## References

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- [10] S. Koppelberg, Applications of σ-filtered Boolean algebras, preprint. Zbl0930.06012
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