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1997 | 154 | 2 | 159-176
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More set-theory around the weak Freese–Nation property

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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)
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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.
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bwmeta1.element.bwnjournal-article-fmv154i2p159bwm
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