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Transactions of the American Mathematical Society

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Good and OK ultrafilters


Author: Alan Dow
Journal: Trans. Amer. Math. Soc. 290 (1985), 145-160
MSC: Primary 54D35; Secondary 04A20, 06E15
DOI: https://doi.org/10.1090/S0002-9947-1985-0787959-4
MathSciNet review: 787959
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Abstract: In this paper we extend Kunen's construction of $ {\alpha ^ + }$-good ultrafilters on $ \mathcal{P}(\alpha )$ to more general algebras, as well as the construction of $ {\alpha ^ + }$-OK ultrafilters. In so doing, we prove the existence of $ ({2^\alpha } \times {\alpha ^ + })$-independent matrices, as defined by Kunen, in these algebras. Some of the topological properties of the Stone spaces of these algebras are then investigated. We find points, for example, in $ U(\alpha )$ which can be regarded as a generalization of weak $ P$-points.


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  • [BF] B. Balcar and F. Franek, Independent families in complete boolean algebras, Trans. Amer. Math. Soc. 274 (1982), 607-618. MR 675069 (83m:06020)
  • [CN] W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer-Verlag, New York, 1974. MR 0396267 (53:135)
  • [DvM] A. Dow and J. van Mill, On nowhere dense $ ccc\;P$-sets, Proc. Amer. Math. Soc. 80 (1980), 697-700. MR 587958 (82a:54032)
  • [EK] R. Engelking and M. Karlowicz, Some theorems of set theory and their topological consequences, Fund. Math. 57 (1965), 275-285. MR 0196693 (33:4880)
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, N. J., 1960. MR 0116199 (22:6994)
  • [Ke1] H. J. Keisler, Good ideals in fields of sets, Ann. of Math. (2) 79 (1964), 338-359. MR 0166105 (29:3383)
  • [Ke2] -, Ideals with prescribed degree of goodness, Ann. of Math. (2) 81 (1965), 112-116. MR 0170828 (30:1063)
  • [K1] K. Kunen, Ultrafilters and independent sets, Trans. Amer. Math. Soc. 172 (1972), 299-306. MR 0314619 (47:3170)
  • [K2] -, Weak $ P$-points in $ \beta N - N$, Proc. Bolyai Janos Soc. Colloq. on Topology (Budapest), 1978, pp. 741-749.
  • [vM1] J. van Mill, Sixteen topological types in $ \beta N - N$, Topology Appl. 13 (1981), 43-57. MR 637426 (83c:54036)
  • [vM2] -, Weak $ P$-points in Čech-Stone remainders, Trans. Amer. Math. Soc. (to appear).
  • [N1] S. Negrepontis, Extensions of continuous functions to $ \beta D$, Indag. Math. 30 (1968), 393-400. MR 0240783 (39:2128)
  • [N2] -, The Stone space of the saturated boolean algebras, Trans. Amer. Math. Soc. 141 (1969), 515-527. MR 0248057 (40:1311)
  • [N3] -, Adequate ultrafilters of special boolean algebras, Trans. Amer. Math. Soc. 174 (1972), 345-367. MR 0313052 (47:1607)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0787959-4
Keywords: Good ultrafilter, independent matrix, $ U(\alpha )$
Article copyright: © Copyright 1985 American Mathematical Society

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