On the relationship between mutual and tight stationarity
Let \(\kappa\) be a singular cardinal and \(\langle \kappa_{\xi}: \xi < \operatorname{cf}(\kappa)\rangle\) a sequence of regular cardinals cofinal in \(\kappa\). A stationary sequence is a sequence \(\vec{\xi} = \langle S_{\xi} : \xi < \operatorname{cf}(\kappa) \rangle\) where \(S_{\xi} \subseteq \kappa_{\xi}\) is stationary in \(\kappa_{\xi}\) for all but boundedly many \(\xi < \operatorname{cf}(\kappa)\). \textit{M. Foreman} and \textit{M. Magidor} [Acta Math. 186, No. 2, 271--300 (2001; Zbl 1017.03022)] introduced two new concepts of stationarity, mutual stationarity and tight stationarity. Here, the authors are interested in the case where \(\operatorname{cf}(\kappa) = \omega\) and where each \(S_{\xi}\) consists of ordinals of some fixed uncountable cofinality \(\eta\). \textit{J. Cummings} et al. [Ann. Pure Appl. Logic 142, No. 1--3, 55--75 (2006; Zbl 1096.03060)] constructed models where \(\langle \omega_n : n<\omega \rangle\) have mutually stationary sequences which are not tightly stationary. The main result of this paper is a forcing construction so that there are mutually stationary but not tightly stationary sequences on every increasing \(\omega\)-sequence of regular cardinals. They show that this property is preserved under a class of Prikry type forcing. They give examples in the Cohen and Prikry models of \(\omega\)-sequences of regular cardinals for which there is a non-tightly stationary sequence of stationary subsets consisting of cofinality \(\omega_1\) ordinals. They show that such stationary sequences are mutually stationary in case of interleaved supercompact cardinals.
scientific article; zbMATH DE number 7358536
scientific article