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Seminormal λ-generated ideals on Pκλ

  • C. A. Johnson (a1)

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In this paper we consider the problem of lifting properties of the Fréchet ideal Iκ = {X ⊆ κ: ∣X∣ < κ} on a regular uncountable cardinal κ, to an analogue about Iκλ, the ideal of not unbounded subsets of Pκλ. With this in mind, in §1 we introduce and study the class of seminormal λ-generated ideals on Pκλ. We shall see that ideals belonging to this class display properties which are clearly analogous to those of the Fréchet ideal on κ (for instance, with regard to saturation, normality and weak selectivity) and yet are closely related to Iκλ. Our results here show that if λ = λ, then many restrictions of Iκλ are weakly selective, nowhere precipitous and, quite suprisingly, seminormal (but nowhere normal). These latter two results suggest the question of whether any restriction of Iκλ can ever be normal. In §2 we prove that if κ is strongly inaccessible, λ = 2λ and NSκλ, the ideal of nonstationary subsets of Pκλ, has a mild selective property, then NSκλA = IκλA for some stationary APκλ.

In [1] Baumgartner showed that if κ is weakly compact and P is the collection of indescribable subsets of κ, then P → (P, κ)2. As a Pκλ analogue of indescribability, Carr (see [3]–[5]) introduced the λ-Shelah property, but was unable to derive the natural Pκλ analogue of Baumgartner's result, (where NShκλ is the normal ideal on Pκλ induced by the λ-Shelah property). In §3 we show that the problem lies in the fact that, as far as we know, NShκλ is not sufficiently distributive, and derive conditions which are sufficient and, in a sense, necessary to yield partitions related to .

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[1]Baumgartner, J., Ineffability properties of cardinals, Infinite and finite sets (Keszthely, 1973), Colloquia Mathematica Societatis János Bolyai, vol. 10, part I, North-Holland, Amsterdam, 1975, pp. 109130.
[2]Baumgartner, J., Taylor, A. and Wagon, S., Structural properties of ideals, Dissertationes Mathematicae/Rozprawy Matematyczne, vol. 197 (1982).
[3]Carr, D., Pκλ generalizations of weak compactness, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 32 (1985), pp. 393401.
[4]Carr, D., Pκλ partition relations, preprint.
[5]Carr, D., The structure of ineffability properties of Pκλ preprint.
[6]Jech, T. and Prikry, K., Ideals over uncountable sets: application of almost disjoint functions and generic ultrapowers, Memoirs of the American Mathematical Society, vol. 18 (1979), no. 2 (214).
[7]Johnson, C. A., Some partition relations for ideals on Pκλ, preprint.
[8]Johnson, C. A., Distributive ideals and partition relations, this Journal, vol. 51 (1986), pp. 617625.
[9]Menas, T., On strong compactness and supercompactness, Annals of Mathematical Logic, vol. 7 (1974), pp. 327359.
[10]Zwicker, W., Pκλ combinatorics. I: Stationary coding sets rationalize the club filter, Axiomatic set theory, Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 243259.
[11]Zwicker, W., Notes on the structural properties of ideals on P κλ, handwritten notes.

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