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2000 | 166 | 1-2 | 153-208
Tytuł artykułu

Cellularity of free products of Boolean algebras (or topologies)

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The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, $θ = (2^{cf(μ)})^+$ and $2^μ = μ^+$ then there are Boolean algebras $\mathbb{B}_1,\mathbb{B}_2$ such that
$c(\mathbb{B}_1) = μ, c(\mathbb{B}_2) < θ but c(\mathbb{B}_1*\mathbb{B}_2)=μ^+$.
Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if $\mathbb{B}$ is a ccc Boolean algebra and $μ^{ℶ_ω} ≤ λ = cf(λ) ≤ 2^μ$ then $\mathbb{B}$ satisfies the λ-Knaster condition (using the "revised GCH theorem").
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Twórcy
  • Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel, shelah@math.huji.ac.il
  • Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, U.S.A.
Bibliografia
  • [1] M. Džamonja and S. Shelah, Universal graphs at successors of singular strong limits.
  • [2] R. Engelking and M. Karłowicz, Some theorems of set theory and their topological consequences, Fund. Math. 57 (1965), 275-285.
  • [3] M. Gitik and S. Shelah, On densities of free products, Topology Appl. 88 (1998), 219-238.
  • [4] A. Hajnal, I. Juhász and S. Shelah, Splitting strongly almost disjoint families, Trans. Amer. Math. Soc. 295 (1986), 369-387.
  • [5] A. Hajnal, I. Juhász and Z. Szentmiklóssy, On the structure of CCC partial orders, Algebra Universalis, to appear.
  • [6] R. B. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229-308.
  • [7] A. Kanamori and M. Magidor, The evolution of large cardinal axioms in set theory, in: Higher Set Theory, Lecture Notes in Math. 669, Springer, 1978, 99-275.
  • [8] J. P. Levinski, M. Magidor and S. Shelah, Chang's conjecture for $ℵ_ω$, Israel J. Math. 69 (1990), 161-172.
  • [9] M. Magidor and S. Shelah, When does almost free imply free? (For groups, transversal etc.), J. Amer. Math. Soc. 7 (1994), 769-830.
  • [10] D. Monk, Cardinal Invariants of Boolean Algebras, Lectures in Mathematics, ETH Zurich, Birkhäuser, Basel, 1990.
  • [11] D. Monk, Cardinal Invariants of Boolean Algebras, Progr. Math., Birkhäuser, Basel, 1996.
  • [12] S. Shelah, Categoricity of an abstract elementary class in two successive cardinals, Israel J. Math. accepted;. math.LO/9805146.
  • [13] S. Shelah, PCF and infinite free subsets, Arch. Math. Logic, accepted; math.LO/9807177.
  • [14] S. Shelah, Remarks on Boolean algebras, Algebra Universalis 11 (1980), 77-89.
  • [15] S. Shelah, Simple unstable theories, Ann. Math. Logic 19 (1980), 177-203.
  • [16] S. Shelah, On saturation for a predicate, Notre Dame J. Formal Logic 22 (1981), 239-248.
  • [17] S. Shelah, Products of regular cardinals and cardinal invariants of products of Boolean algebras, Israel J. Math. 70 (1990), 129-187.
  • [18] S. Shelah, Advances in Cardinal Arithmetic, in: Finite and Infinite Combinatorics in Sets and Logic, N. W. Sauer et al. (eds.), Kluwer, 1993, 355-383.
  • [19] S. Shelah, More on cardinal arithmetic, Arch. Math. Logic 32 (1993), 399-428.
  • [20] S. Shelah, $ℵ_{ω + 1}$ has a Jonsson algebra, Chapter II of [20].
  • [21] S. Shelah, Basic: Cofinalities of small reduced products, Chapter I of [20].
  • [22] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, 1994.
  • [23] S. Shelah, Further cardinal arithmetic, Israel J. Math. 95 (1996), 61-114; math. LO/9610226.
  • [24] S. Shelah, Colouring and non-productivity of $ℵ_2$-cc, Ann. Pure Appl. Logic 84 (1997), 153-174; math.LO/9609218.
  • [25] S. Shelah, The Generalized Continuum Hypothesis revisited, Israel J. Math. 116 (2000), 285-321; math.LO/9809200.
  • [26] R. M.Solovay, Strongly compact cardinals and the GCH, in: Proc. Tarski Symposium (Berkeley, 1971), Proc. Sympos. Pure Math. 25, Amer. Math. Soc., 1974, 365-372.
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Bibliografia
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bwmeta1.element.bwnjournal-article-fmv166i1p153bwm
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