Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Independent families in complete Boolean algebras
HTML articles powered by AMS MathViewer

by B. Balcar and F. Franěk PDF
Trans. Amer. Math. Soc. 274 (1982), 607-618 Request permission

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 $2^{|B|}$.
References
    M. Benda, Reduced products, filters and Boolean ultrapowers, Ph.D. Thesis, University of Wisconsin, 1970.
  • Bohuslav Balcar and Peter Vojtás, Refining systems on Boolean algebras, Set theory and hierarchy theory, V (Proc. Third Conf., Bierutowice, 1976) Lecture Notes in Math., Vol. 619, Springer, Berlin, 1977, pp. 45–58. MR 0498304
    • A. Blaszczyk, On mappings of extremally disconnected compact spaces onto Cantor cubes, Proc. Colloq. Topology, Budapest, 1978. L. Bukovsky, Boolean ultrapowers and elementary equivalence, Abstract of 4th ICLMPS Bucharest, 1971, pp. 13-14.
  • W. W. Comfort and Anthony W. Hager, Cardinality of ${\mathfrak {k}}$-complete Boolean algebras, Pacific J. Math. 40 (1972), 541–545. MR 307997
  • W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974. MR 0396267
    • C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
  • 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
  • R. Engelking and M. Karłowicz, Some theorems of set theory and their topological consequences, Fund. Math. 57 (1965), 275–285. MR 196693, DOI 10.4064/fm-57-3-275-285
  • P. Erdös and A. Tarski, On families of mutually exclusive sets, Ann. of Math. (2) 44 (1943), 315–329. MR 8249, DOI 10.2307/1968767
  • 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 Russian summary). MR 233343
    • F. Hausdorff, Uber zwei Sätze von G. Fichtenholz und L. Kantorovich, Studia Math. 6 (1936), 18-19.
  • Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
  • 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
  • H. Jerome Keisler, Good ideals in fields of sets, Ann. of Math. (2) 79 (1964), 338–359. MR 166105, DOI 10.2307/1970549
  • H. Jerome Keisler, Ultraproducts and saturated models, Nederl. Akad. Wetensch. Proc. Ser. A 67 = Indag. Math. 26 (1964), 178–186. MR 0168483
    • J. Ketonen, Everything you wanted to know about ultrafilters but were afraid to ask, Ph.D. Thesis, University of Wisconsin, 1971. S. Koppelberg, Free subalgebras of complete Boolean algebras, Notices Amer. Math. Soc. 20 (1973), A-418.
  • Kenneth Kunen, Ultrafilters and independent sets, Trans. Amer. Math. Soc. 172 (1972), 299–306. MR 314619, DOI 10.1090/S0002-9947-1972-0314619-7
  • K. Kunen, Weak $P$-points in $\textbf {N}^{\ast }$, 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
  • 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 245490, DOI 10.1002/malq.19690150705
  • Richard Mansfield, The theory of Boolean ultrapowers, Ann. Math. Logic 2 (1970/71), no. 3, 297–323. MR 300887, DOI 10.1016/0003-4843(71)90017-9
  • J. Donald Monk, On free subalgebras of complete Boolean algebras, Arch. Math. (Basel) 29 (1977), no. 2, 113–115. MR 472626, DOI 10.1007/BF01220382
    • V. I. Ponomarev and L. B. Shapiro, Absolutes of topological spaces and their continuous maps, Russian Math. Surveys 31 (1976), no. 5, 138-154.
  • Bedřich Pospíšil, Remark on bicompact spaces, Ann. of Math. (2) 38 (1937), no. 4, 845–846. MR 1503375, DOI 10.2307/1968840
  • Bedřich Pospíšil, On bicompact spaces, Publ. Fac. Sci. Univ. Masaryk 1939 (1939), no. 270, 16. MR 1454
  • D. A. Vladimirov, Bulevy algebry, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0263713
Similar Articles
Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 274 (1982), 607-618
  • MSC: Primary 06E10; Secondary 03G05, 04A20, 54A20, 54G05
  • DOI: https://doi.org/10.1090/S0002-9947-1982-0675069-3
  • MathSciNet review: 675069