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## Fundamenta Mathematicae

1997 | 154 | 2 | 159-176
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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)
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
159-176
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-04-29
poprawiono
1996-05-24
Twórcy
autor
• Institut für Mathematik II, Freie Universität Berlin, 14195 Berlin, Germany
autor
• Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary
Bibliografia
• [1] T. Bartoszyński and H. Judah, Set Theory: on the structure of the real line, A K Peters, 1995.
• [2] S. Ben-David and M. Magidor, The weak □ is really weaker than full □, J. Symbolic Logic 51 (1986), 1029-1033.
• [3] M. Foreman and M. Magidor, A very weak square principle, preprint.
• [4] M. Foreman, M. Magidor and S. Shelah, Martin's maximum, saturated ideals, and non-regular ultrafilters I, Ann. of Math. (2) 127 (1988), 1-47.
• [5] R. Freese and J. B. Nation, Projective lattices, Pacific J. Math. 75 (1978), 93-106.
• [6] S. Fuchino, S. Koppelberg and S. Shelah, Partial orderings with the weak Freese-Nation property, Ann. Pure Appl. Logic 80 (1996), 35-54.
• [7] S. Fuchino, S. Koppelberg and S. Shelah, A game on partial orderings, Topology Appl. 74 (1996), 141-148.
• [8] L. Heindorf and L. B. Shapiro, Nearly Projective Boolean Algebras, Lecture Notes in Math. 1596, Springer, 1994.
• [9] R. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229-308.
• [10] S. Koppelberg, Applications of σ-filtered Boolean algebras, preprint.
• [11] S. Koppelberg and S. Shelah, Subalgebras of the Cohen algebra do not have to be Cohen, preprint.
• [12] K. Kunen, Set Theory, North-Holland, 1980.
• [13] J.-P. Levinski, M. Magidor and S. Shelah, On Chang's conjecture for $ℵ_ω$, Israel J. Math. 69 (1990), 161-172.
Typ dokumentu
Bibliografia
Identyfikatory