## Variations of cub filter on $\mathcal {P}_ \kappa \lambda$

HTML articles powered by AMS MathViewer

- by Yo Matsubara
- Proc. Amer. Math. Soc.
**102**(1988), 1009-1017 - DOI: https://doi.org/10.1090/S0002-9939-1988-0934884-X
- PDF | Request permission

## Abstract:

In his 1973 paper, Jech extended the notion of cub and stationary sets to such sets in ${\mathcal {P}_\kappa }\lambda$ and showed that many of their properties are preserved. We study variations of cub filters in this paper. We make use of the partition property (a large cardinal hypothesis) to investigate the properties of these filters. In the last section we investigate the relation of our filter to supercompact filters on ${\mathcal {P}_{{\aleph _1}}}\lambda$ under the Axiom of Determinacy. This motivates the formulation of a certain infinitary partition property, and this property implies the $\lambda$-supercompactness of ${\aleph _1}$.## References

- Stewart Baldwin,
*The consistency strength of certain stationary subsets of ${\scr P}_{\kappa }\lambda$*, Proc. Amer. Math. Soc.**92**(1984), no.Β 1, 90β92. MR**749898**, DOI 10.1090/S0002-9939-1984-0749898-9 - Howard Becker,
*$\textrm {AD}$ and the supercompactness of $\aleph _{1}$*, J. Symbolic Logic**46**(1981), no.Β 4, 822β842. MR**641495**, DOI 10.2307/2273231 - Leo A. Harrington and Alexander S. Kechris,
*On the determinacy of games on ordinals*, Ann. Math. Logic**20**(1981), no.Β 2, 109β154. MR**622782**, DOI 10.1016/0003-4843(81)90001-2 - Thomas J. Jech,
*Some combinatorial problems concerning uncountable cardinals*, Ann. Math. Logic**5**(1972/73), 165β198. MR**325397**, DOI 10.1016/0003-4843(73)90014-4 - E. M. Kleinberg,
*Strong partition properties for infinite cardinals*, J. Symbolic Logic**35**(1970), 410β428. MR**309734**, DOI 10.2307/2270698 - David W. Kueker,
*Countable approximations and LΓΆwenheim-Skolem theorems*, Ann. Math. Logic**11**(1977), no.Β 1, 57β103. MR**457191**, DOI 10.1016/0003-4843(77)90010-9 - M. Magidor,
*Combinatorial characterization of supercompact cardinals*, Proc. Amer. Math. Soc.**42**(1974), 279β285. MR**327518**, DOI 10.1090/S0002-9939-1974-0327518-9 - Yo Matsubara,
*Menasβ conjecture and generic ultrapowers*, Ann. Pure Appl. Logic**36**(1987), no.Β 3, 225β234. MR**915899**, DOI 10.1016/0168-0072(87)90018-2 - Telis K. Menas,
*On strong compactness and supercompactness*, Ann. Math. Logic**7**(1974/75), 327β359. MR**357121**, DOI 10.1016/0003-4843(75)90009-1 - Telis K. Menas,
*A combinatorial property of $p_{k}\lambda$*, J. Symbolic Logic**41**(1976), no.Β 1, 225β234. MR**409186**, DOI 10.2307/2272962 - Yiannis N. Moschovakis,
*Descriptive set theory*, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR**561709**
W. Hugh Woodin, AD - William S. Zwicker,
*$P_\kappa \lambda$ combinatorics. II. The RK ordering beneath a supercompact measure*, J. Symbolic Logic**51**(1986), no.Β 3, 604β616. MR**853843**, DOI 10.2307/2274017

*and the uniqueness of the supercompact measures on*${\mathcal {P}_{{\omega _1}}}\lambda$, Cabal Seminar 79-81, Lecture Notes in Math., vol. 1019, Springer-Verlag, 1983.

## Bibliographic Information

- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**102**(1988), 1009-1017 - MSC: Primary 03E05; Secondary 03E55, 03E60, 04A20
- DOI: https://doi.org/10.1090/S0002-9939-1988-0934884-X
- MathSciNet review: 934884