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June 2013 On skinny stationary subsets of $\mathcal {P}_\kappa \lambda $
Yo Matsubara, Toschimichi Usuba
J. Symbolic Logic 78(2): 667-680 (June 2013). DOI: 10.2178/jsl.7802180

Abstract

We introduce the notion of skinniness for subsets of $\mathcal{P}_\kappa \lambda$ and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or $2^\lambda$-saturation of $\mathrm{NS}_{\kappa\lambda}\mid X$, where $\mathrm{NS}_{\kappa\lambda}$ denotes the non-stationary ideal over $\mathcal{P}_\kappa \lambda$, implies the existence of a skinny stationary subset of $X$. We also show that if $\lambda$ is a singular cardinal, then there is no skinnier stationary subset of $\mathcal{P}_\kappa \lambda$. Furthermore, if $\lambda$ is a strong limit singular cardinal, there is no skinny stationary subset of $\mathcal{P}_\kappa \lambda$. Combining these results, we show that if $\lambda$ is a strong limit singular cardinal, then $\mathrm{NS}_{\kappa\lambda}\mid X$ can satisfy neither precipitousness nor $2^\lambda$-saturation for every stationary $X \subseteq \mathcal{P}_\kappa \lambda$. We also indicate that $\diamondsuit_\lambda(E^{\lambda}_{<\kappa})$, where $E^{\lambda}_{<\kappa} \stackrel{\mathrm{def}}{=} \{\alpha < \lambda \mid \mathrm{cf}(\alpha) < \kappa\}$, is equivalent to the existence of a skinnier (or skinniest) stationarysubset of $\mathcal{P}_\kappa \lambda$ under some cardinal arithmeticalhypotheses.

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Yo Matsubara. Toschimichi Usuba. "On skinny stationary subsets of $\mathcal {P}_\kappa \lambda $." J. Symbolic Logic 78 (2) 667 - 680, June 2013. https://doi.org/10.2178/jsl.7802180

Information

Published: June 2013
First available in Project Euclid: 15 May 2013

zbMATH: 1275.03142
MathSciNet: MR3145202
Digital Object Identifier: 10.2178/jsl.7802180

Subjects:
Primary: 03E35, 03E55

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 2 • June 2013
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