More set-theory around the weak Freese–Nation property

Sakaé Fuchino; Lajos Soukup

Fundamenta Mathematicae (1997)

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

Abstract

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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)

How to cite

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Fuchino, 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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  8. [8] L. Heindorf and L. B. Shapiro, Nearly Projective Boolean Algebras, Lecture Notes in Math. 1596, Springer, 1994. 
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  10. [10] S. Koppelberg, Applications of σ-filtered Boolean algebras, preprint. Zbl0930.06012
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  12. [12] K. Kunen, Set Theory, North-Holland, 1980. 
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