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The Rudin-Blass ordering of ultrafilters

  • Claude Laflamme (a1) and Jian-Ping Zhu (a2)
  • DOI:
  • Published online: 01 March 2014

We discuss the finite-to-one Rudin-Keisler ordering of ultrafilters on the natural numbers, which we baptize the Rudin-Blass ordering in honour of Professor Andreas Blass who worked extensively in the area.

We develop and summarize many of its properties in relation to its bounding and dominating numbers, directedness, and provide applications to continuum theory. In particular, we prove in ZFC alone that there exists an ultrafilter with no Q-point below in the Rudin-Blass ordering.

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[2]A. Blass , Near coherence of filters I: cofinal equivalence of models of arithmetic, Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 579581.

[3]A. Blass , Near coherence of filters II: the Stone-Cech remainder of half-line, Transactions of the American Mathematical Society, vol. 300 (1987), pp. 557581.

[4]A. Blass and S. Shelah , There may be simple and points and the Rudin-Keisler ordering may be downward directed, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 213243.

[5]M. Canjar , Cofinalities of countable ultraproducts: the existence theorem, Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 539542.

[6]W. Comfort and S. Negrepontis , The theory of ultrafilters, Springer-Verlag, Berlin, 1974.

[10]C. Laflamme , Forcing with filters and complete combinatorics, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 125163.

[12]J.-P. Zhu , Continua in ℝ*, Topology audits Applications, vol. 50 (1993), pp. 183197.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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