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On skinny stationary subsets of

Published online by Cambridge University Press:  12 March 2014

Yo Matsubara  and
Toshimichi Usuba
Yo Matsubara
Affiliation:
Graduate School of Information Science, Nagoya University, Furo-Cho, Chikusa-Ku, Nagoya, 464-8601, Japan, E-mail:yom@math.nagoya-u.ac.jp
Toshimichi Usuba
Affiliation:
Institute for Advanced Research, Nagoya University, Furo-Cho, Chikusa-Ku, Nagoya, 464-8601, Japan, E-mail:usuba@math.cm.is.nagoya-u.ac.jp
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Abstract

We introduce the notion of skinniness for subsets of and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or 2λ-saturation of NSκλ ∣ X, where NSκλ denotes the non-stationary ideal over , implies the existence of a skinny stationary subset of X. We also show that if λ is a singular cardinal, then there is no skinnier stationary subset of . Furthermore, if λ is a strong limit singular cardinal, there is no skinny stationary subset of . Combining these results, we show that if λ is a strong limit singular cardinal, then NSκλ ∣ X can satisfy neither precipitousness nor 2λ-saturation for every stationary X ⊆ . We also indicate that , where , is equivalent to the existence of a skinnier (or skinniest) stationary subset of under some cardinal arithmetical hypotheses.

Keywords

non-stationary idealprecipitous idealsaturated ideal
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 2 , June 2013 , pp. 667 - 680
Copyright
Copyright © Association for Symbolic Logic 2013

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References

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