Abstract
An ideal J on \([\lambda ]^{<\kappa }\) is said to be \([\delta ]^{<\theta }\)-normal, where \(\delta \) is an ordinal less than or equal to \(\lambda \), and \(\theta \) a cardinal less than or equal to \(\kappa \), if given \(B_e \in J\) for \(e \in [\delta ]^{<\theta }\), the set of all \(a \in [\lambda ]^{<\kappa }\) such that \(a \in B_e\) for some \(e \in [a \cap \delta ]^{< \vert a \cap \theta \vert }\) lies in J. We give necessary and sufficient conditions for the existence of such ideals and describe the smallest one, denoted by \(NS_{\kappa ,\lambda }^{[\delta ]^{<\theta }}\). We compute the cofinality of \(NS_{\kappa ,\lambda }^{[\delta ]^{<\theta }}\).
References
Abe, Y.: A hierarchy of filters smaller than \(CF_{\kappa \lambda }\). Arch. Math. Log. 36, 385–397 (1997)
Carr, D.M.: The minimal normal filter on \(P_\kappa \lambda \). Proc. Am. Math. Soc. 86, 316–320 (1982)
Carr, D.M., Levinski, J.P., Pelletier, D.H.: On the existence of strongly normal ideals on \(P_\kappa \lambda \). Arch. Math. Log. 30, 59–72 (1990)
Carr, D.M., Pelletier, D.H.: Towards a structure theory for ideals on \(P_\kappa \lambda \). In: Steprāns J., Watson S. (eds.) Set Theory and its Applications, Lecture Notes in Mathematics, vol. 1401, pp. 41–54. Springer, Berlin (1989)
Donder, H.D., Matet, P.: Two cardinal versions of diamond. Isr. J. Math. 83, 1–43 (1993)
Džamonja, M.: On \(P_\kappa \lambda \)-combinatorics using a third cardinal. Radovi Matematički 9, 145–155 (1999)
Erdös, P., Hajnal, A., Máté, A., Rado, R.: Combinatorial Set Theory: Partition Relations for Cardinals, Studies in Logic and the Foundations of Mathematics, vol. 106. North-Holland, Amsterdam (1984)
Feng, Q.: An ideal characterization of Mahlo cardinals. J. Symb. Log. 54, 467–473 (1989)
Holz, M., Steffens, K., Weitz, E.: Introduction to Cardinal Arithmetic, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Basel (1999)
Jech, T.J.: The closed unbounded filter on \(P_\kappa (\lambda )\). Not. Am. Math. Soc. 18, 663 (1971)
Kunen, K.: Set Theory, Studies in Logic and the Foundations of Mathematics, vol. 102. North-Holland, Amsterdam (1980)
Landver, A.: Singular Baire Numbers and Related Topics. Ph.D. Thesis, University of Wisconsin, Madison (1990)
Matet, P.: Un principe combinatoire en relation avec l’ultranormalité des idéaux. C. R. Acad. Sci. Paris Sér. I 307, 61–62 (1988)
Matet, P.: Concerning stationary subsets of \([\lambda ]^ {<\kappa } \). In: Steprāns J., Watson S. (eds.) Set Theory and Its Applications , Lecture Notes in Mathematics, vol. 1401, pp. 119–127. Springer, Berlin (1989)
Matet, P.: Partition relations for \(\kappa \)-normal ideals on \(P_\kappa (\lambda )\). Ann. Pure Appl. Log. 121, 89–111 (2003)
Matet, P.: Covering for category and combinatorics on \(P_\kappa (\lambda )\). J. Math. Soc. Jpn. 58, 153–181 (2006)
Matet, P.: Large cardinals and covering numbers. Fundam. Math. 205, 45–75 (2009)
Matet, P., Péan, C., Shelah, S.: Cofinality of normal ideals on \(P_\kappa (\lambda )\) II. Isr. J. Math. 121, 89–111 (2003)
Menas, T.K.: On strong compactness and supercompactness. Ann. Math. Log. 7, 327–359 (1974)
Shelah, S.: Cardinal Arithmetic, Oxford Logic Guides, vol. 29. Oxford University Press, Oxford (1994)
Shelah, S.: The generalized continuum hypothesis revisited. Isr. J. Math. 116, 285–321 (2000)
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Cédric Péan: Some of the material in this paper originally appeared as part of the author’s doctoral dissertation completed at the Université de Caen, 1998.
Saharon Shelah: Partially supported by the Israel Science Foundation. Publication 713.
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Matet, P., Péan, C. & Shelah, S. Cofinality of normal ideals on \([\lambda ]^{<\kappa }\) I. Arch. Math. Logic 55, 799–834 (2016). https://doi.org/10.1007/s00153-016-0496-5
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DOI: https://doi.org/10.1007/s00153-016-0496-5