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

Published online by Cambridge University Press:  12 March 2014

Claude Laflamme
Claude Laflamme*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1, Canada
*
Department of Mathematics, York University, 4700 Keele Street, North York, Ontario M3J 1P3, Canada
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 2 , June 1990 , pp. 449 - 456
Copyright
Copyright © Association for Symbolic Logic 1990

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References

REFERENCES

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