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Some filters of partitions

Published online by Cambridge University Press:  12 March 2014

Pierre Matet
Pierre Matet*
Affiliation:
Institut für Mathematik II, Freie Universität Berlin, 1000 Berlin 33, West Germany
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Extract

§0. Introduction. We started our study of filters of partitions in [15]. We shall here restrict ourselves to the consideration of filters on (ω)ω, the set of all infinite partitions of ω. §1 is an attempt to elucidate the connection between filters on (ω)ω and filters over ω. Given a filter H over ω, we define two filters FH and GH on (ω)ω, and we characterize p-points, rare ultrafilters and Ramsey ultrafilters in terms of properties of the associated filters of partitions.

The remainder of the paper is devoted to the study of those filters that can be associated with Hindman's theorem and its extensions. Let us introduce some notation. Suppose * is an associative operation on ω, and let a subset A of ω and an ordinal α with 0 < α ≤ ω be given.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 2 , June 1988 , pp. 540 - 553
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[1] Blass, A., Ultrafilters related to Hindman's finite-unions theorem and its extensions, Logic and combinatorics (Simpson, S., editor), Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, Rhode Island, 1987, pp. 89–124.CrossRefGoogle Scholar
[2] Blass, A. and Hindman, N., On strongly summable ultrafilters and union uttrafilters, Transactions of the American Mathematical Society, vol. 304 (1987), pp. 83–99.CrossRefGoogle Scholar
[3] Booth, D., Ultrafilters over a countable set, Annals of Mathematical Logic, vol. 2 (1970), pp. 1–24.CrossRefGoogle Scholar
[4] Carlson, T. J. and Simpson, S. G., A dual form of Ramsey's theorem, Advances in Mathematics, vol. 53 (1984), pp. 265–290.CrossRefGoogle Scholar
[5] Choquet, G., Deux classes remarquables d'ultrafiltres sur N, Bulletin des Sciences Mathématiques, sèr. 2, vol. 92 (1968), pp. 143–153.Google Scholar
[6] Comfort, W. W., Ultrafilters: some old and some new results, Bulletin of the American Mathematical Society, vol. 83 (1977), pp. 417–455.CrossRefGoogle Scholar
[7] Hindman, N., The existence of certain ultrafilters on N and a conjecture of Graham and Rothschild, Proceedings of the American Mathematical Society, vol. 36 (1972), pp. 341–346.Google Scholar
[8] Hindman, N., Finite sums from sequences within cells of a partition of N, Journal of Combinatorial Theory, Series A, vol. 17 (1974), pp. 1–11.CrossRefGoogle Scholar
[9] Hindman, N., Ultrafilters and combinatorial set theory, Number theory, Carbondale 1979, Lecture Notes in Mathematics, vol. 751, Springer-Verlag, Berlin, 1979, pp. 119–184.CrossRefGoogle Scholar
[10] Hindman, N., Summable ultrafilters and finite sums, Logic and combinatorics (Simpson, S., editor), Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, Rhode Island, 1987, pp. 263–274.CrossRefGoogle Scholar
[11] Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[12] Kunen, K., Some points in βN, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 80 (1976), pp. 385–398.CrossRefGoogle Scholar
[13] Louveau, A., Une méthode topologique pour l'étude de la propriété de Ramsey, Israel Journal of Mathematics, vol. 23 (1976), pp. 97–115.CrossRefGoogle Scholar
[14] Martin, D. A. and Solovay, R. M., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143–178.CrossRefGoogle Scholar
[15] Matet, P., Partitions and filters, this Journal, vol. 51 (1986), pp. 12–21.Google Scholar
[16] Mathias, A. R. D., Happy families, Annals of Mathematical Logic, vol. 12 (1977), pp. 59–111.CrossRefGoogle Scholar
[17] Milliken, K., Ramsey's theorem with sums or unions, Journal of Combinatorial Theory, Series A, vol. 18 (1975), pp. 276–290.CrossRefGoogle Scholar
[18] Mokobodzki, G., Ultrafiltres rapides sur N. Construction d'une densité relative de deux potentiels comparables , Séminaire de théorie du potentiel (Brelot, M., Choquet, G. et Deny, J.), 1967/1968, Secrétariat Mathématique, Paris, 1969, Exposé 12.Google Scholar
[19] Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[20] Stern, J., Regularity properties of definable sets of reals, Annals of Pure and Applied Logic, vol. 29 (1985), pp. 289–324.CrossRefGoogle Scholar
[21] Taylor, A., A canonical partition relation for finite subsets of ω, Journal of Combinatorial Theory, Series A, vol. 21 (1976), pp. 137–146.CrossRefGoogle Scholar