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Upward directedness of the Rudin-Keisler ordering of p-points

  • Claude Laflamme (a1)

It has been proven by Blass [1973] that any two P-points which have a P-point as a common upper bound in the Rudin-Keisler (RK) ordering necessarily have a common lower bound (necessarily a P-point). Hence two nonisomorphic Ramsey ultrafilters have neither a common lower nor a common upper bound which is a P-point. So in a model of CH for example (or MA, P(c),…), the RK ordering restricted to P-points is neither upward nor downward directed, since it is well known that nonisomorphic Ramsey ultrafilters exist in such models. On the other hand, we will see that in the model for “near coherence of filters” (NCF) produced by Blass and Shelah [1985], the RK ordering of P-points is upward, hence downward directed. This shows that the question of directedness of the RK ordering of P-points, upward or downward, cannot be decided in ZFC.

There is a related question, asked by Blass in [1973], whether two P-points which have a common lower bound necessarily have a common upper bound which is a P-point. Our main result establishes the independence of this statement relative to ZFC. Its consistency will follow as soon as we show that the RK ordering of P-points is upward directed in the NCF model mentioned above, which we do in §2. But its independence will require a new construction, and will be given in §3.

Corresponding author
Department of Mathematics, York University, 4700 Keele Street, North York, Ontario M3J 1P3, Canada
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Blass, A. [1973], The Rudin-Keisler ordering of P-points, Transactions of the American Mathematical Society, vol. 179, pp. 145166.
Blass, A. [1981], Some initial segments of the Rudin-Keisler ordering, this Journal, vol. 46, pp. 147157.
Blass, A. and Shelah, S. [1987], There may be simple and -points and the Rudin-Keisler ordering may be downward directed, Annals of Pure and Applied Logic, vol. 33, pp. 213243.
Booth, D. [1970], Ultrafilters on a countable set, Annals of Mathematical Logic, vol. 2, pp. 124.
Ketonen, J. [1976], On the existence of P-points in the Stone-Čech compactification of the integers, Fundamenta Mathematicae, vol. 92, pp. 9194.
Kunen, K. [1976], Some points in βN, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 80, pp. 385398.
Kunen, K. [1980], Set theory: an introduction to independence proofs, North-Holland, Amsterdam.
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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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