Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Normal filters generated by a family of sets


Authors: Carlos Augusto Di Prisco and Carlos Enrique Uzcategui
Journal: Proc. Amer. Math. Soc. 101 (1987), 513-518
MSC: Primary 04A20; Secondary 03E05
DOI: https://doi.org/10.1090/S0002-9939-1987-0908659-0
MathSciNet review: 908659
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study normal filters on the set spaces $ \lambda ,{\mathcal{P}_\kappa }\left( \lambda \right),{\left[ \lambda \right]^\kappa }$, and $ {\left( \lambda \right)^\kappa }$. We characterize the least normal $ \gamma $-complete filter containing a given $ \gamma $-complete filter for $ \gamma \geq {\omega _1}$. If $ \mathcal{F}$ is a $ {\omega _1}$-complete filter on any of the set spaces mentioned, the least $ {\omega _1}$-complete normal filter containing it is the filter generated by the sets $ \left\{ {x \in E\left\vert {{\alpha _1}, \ldots ,{\alpha _n} \in x \to x \in f\left( {{\alpha _1}, \ldots ,{\alpha _n}} \right)} \right.} \right\}$ where $ f:{\lambda ^{ < \omega }} \to \mathcal{F}$.


References [Enhancements On Off] (What's this?)

  • [BTW] J. E. Baumgartner, A. D. Taylor and S. Wagon, Structural properties of ideals, Dissertationes Math. 197 (1982). MR 687276 (84f:04007)
  • [C] D. Carr, The minimal normal filter on $ {\mathcal{P}_\kappa }\left( \lambda \right)$, Proc. Amer. Math. Soc. 86 (1982), 316-320. MR 667297 (84h:03109)
  • [DP.MI] C. A. Di Prisco and W. Marek, A filter on $ {\left[ \lambda \right]^k}$, Proc. Amer. Math. Soc. 90 (1984), 591-598. MR 733412 (85g:03072)
  • [DP.M2] -, On the space $ {\left( \lambda \right)^\kappa }$, Methods in Mathematical Logic (C. A. Di Prisco, ed.), Lecture Notes in Math., vol. 1130, Springer-Verlag, 1985, pp. 151-156. MR 799039 (87b:03104)
  • [F] G. Fodor, Eine Bemerkung zur Theorie de regressiven Funktionen, Acta Sei. Math. (Szeged) 17 (1956), 139-142. MR 0082450 (18:551d)
  • [Je] Thomas Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1973), 168-198. MR 0325397 (48:3744)
  • [Me] T. K. Menas, On strong compactness and supercompactness, Ann. Math. Logic 7 (1974), 327359. MR 0357121 (50:9589)
  • [SRK] R. Solovay, W. Reinhardt and A. Kanamori, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), 73-116. MR 482431 (80h:03072)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 04A20, 03E05

Retrieve articles in all journals with MSC: 04A20, 03E05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0908659-0
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society