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Transactions of the American Mathematical Society

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Independent families in complete Boolean algebras

Authors: B. Balcar and F. Franěk
Journal: Trans. Amer. Math. Soc. 274 (1982), 607-618
MSC: Primary 06E10; Secondary 03G05, 04A20, 54A20, 54G05
MathSciNet review: 675069
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Abstract: We present a proof (without any set-theoretical assumptions) that every infinite complete Boolean algebra includes a free subalgebra of the same cardinality. It follows that the set of all ultrafilters on an infinite complete Boolean algebra $ B$ has power $ 2^{\vert B\vert}$.

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  • [Be] M. Benda, Reduced products, filters and Boolean ultrapowers, Ph.D. Thesis, University of Wisconsin, 1970.
  • [B, V] Bohuslav Balcar and Peter Vojtás, Refining systems on Boolean algebras, Set theory and hierarchy theory, V (Proc. Third Conf., Bierutowice, 1976), Springer, Berlin, 1977, pp. 45–58. Lecture Notes in Math., Vol. 619. MR 0498304
  • [Bla] A. Blaszczyk, On mappings of extremally disconnected compact spaces onto Cantor cubes, Proc. Colloq. Topology, Budapest, 1978.
  • [Bu] L. Bukovsky, Boolean ultrapowers and elementary equivalence, Abstract of 4th ICLMPS Bucharest, 1971, pp. 13-14.
  • [C, H] W. W. Comfort and Anthony W. Hager, Cardinality of 𝔨-complete Boolean algebras, Pacific J. Math. 40 (1972), 541–545. MR 0307997
  • [C, N] W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 211. MR 0396267
  • [Ch, K] C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
  • [E] B. A. Efimov, Extremally disconnected bicompacta and absolutes (on the occasion of the one hundredth anniversary of the birth of Felix Hausdorff), Trudy Moskov. Mat. Obšč. 23 (1970), 235–276 (Russian). MR 0418016
  • [E, K] R. Engelking and M. Karłowicz, Some theorems of set theory and their topological consequences, Fund. Math. 57 (1965), 275–285. MR 0196693
  • [E, T] P. Erdös and A. Tarski, On families of mutually exclusive sets, Ann. of Math. (2) 44 (1943), 315–329. MR 0008249
  • [F] Z. Frolík, Fixed points of maps of extremally disconnected spaces and complete Boolean algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 269–275 (English, with Loose Russian summary). MR 0233343
  • [H] F. Hausdorff, Uber zwei Sätze von G. Fichtenholz und L. Kantorovich, Studia Math. 6 (1936), 18-19.
  • [J] Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. MR 506523
  • [K] S. V. Kisljakov, Free subalgebras of complete Boolean algebras, and spaces of continuous functions, Sibirsk. Mat. Ž. 14 (1973), 569–581, 693–694 (Russian). MR 0340138
  • [K1] H. Jerome Keisler, Good ideals in fields of sets, Ann. of Math. (2) 79 (1964), 338–359. MR 0166105
  • [K2] H. Jerome Keisler, Ultraproducts and saturated models, Nederl. Akad. Wetensch. Proc. Ser. A 67 = Indag. Math. 26 (1964), 178–186. MR 0168483
  • [Ke] J. Ketonen, Everything you wanted to know about ultrafilters but were afraid to ask, Ph.D. Thesis, University of Wisconsin, 1971.
  • [Ko] S. Koppelberg, Free subalgebras of complete Boolean algebras, Notices Amer. Math. Soc. 20 (1973), A-418.
  • [Ku1] Kenneth Kunen, Ultrafilters and independent sets, Trans. Amer. Math. Soc. 172 (1972), 299–306. MR 0314619, 10.1090/S0002-9947-1972-0314619-7
  • [Ku2] K. Kunen, Weak 𝑃-points in 𝑁*, Topology, Vol. II (Proc. Fourth Colloq., Budapest, 1978) Colloq. Math. Soc. János Bolyai, vol. 23, North-Holland, Amsterdam-New York, 1980, pp. 741–749. MR 588822
  • [Ma] D. C. Makinson, On the number of ultrafilters of an infinite Boolean algebra, Z. Math. Logik Grundlagen Math. 15 (1969), 121–122 (English, with French summary). MR 0245490
  • [Mn] Richard Mansfield, The theory of Boolean ultrapowers, Ann. Math. Logic 2 (1970/71), no. 3, 297–323. MR 0300887
  • [M] J. Donald Monk, On free subalgebras of complete Boolean algebras, Arch. Math. (Basel) 29 (1977), no. 2, 113–115. MR 0472626
  • [P, S] V. I. Ponomarev and L. B. Shapiro, Absolutes of topological spaces and their continuous maps, Russian Math. Surveys 31 (1976), no. 5, 138-154.
  • [P1] Bedřich Pospíšil, Remark on bicompact spaces, Ann. of Math. (2) 38 (1937), no. 4, 845–846. MR 1503375, 10.2307/1968840
  • [P2] Bedřich Pospíšil, On bicompact spaces, Publ. Fac. Sci. Univ. Masaryk 1939 (1939), no. 270, 16. MR 0001454
  • [V] D. A. Vladimirov, Bulevy algebry, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0263713

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Article copyright: © Copyright 1982 American Mathematical Society