We show that if κ is an infinite successor cardinal, and λ > κ a cardinal of cofinality less than κ satisfying certain conditions, then no ideal on Pκ is weakly λ+-saturated. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Assuming Jensen's principle ◊+ we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and Todorčević.

Given a regular uncountable cardinal κ and a cardinal λ > κ of cofinality ω, we show that the restriction of the non-stationary ideal on Pκ to the set of all a with equation image is not λ++-saturated . We actually prove the stronger result that there is equation image with |Q| = λ++ such that A∩B is a non-cofinal subset of Pκ for any two distinct members A, B of Q, where NGκ, λ denotes the game ideal on Pκ. (...) We also remark that for κ > ω1, adding λ+3 Cohen subsets of ω1 to equation image makes NGκ, λ λ+3-saturated. (shrink)

The structure Dense /nwd and gaps in analytic quotients of equation image have been studied in the literature 2, 3, 1. We prove that the structures Dense /nwd and equation image have gaps of type equation image, and there are no -gaps for equation image, where equation image is the additivity number of the meager ideal. We also prove the existence of -gaps in these structures. Finally we characterize the cofinality of the meager ideal equation image using families of sets (...) in Dense /nwd. (shrink)

The structure \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{Dense}(\mathbb {Q})/\mathbf {nwd}$\end{document} and gaps in analytic quotients of \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathscr {P}(\omega )$\end{document} have been studied in the literature 2, 3, 1. We prove that the structures \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathrm{Dense} (\mathbb {Q})/\mathbf {nwd}$\end{document} and \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathscr {P}(\mathbb {Q})/\mathbf {nwd}$\end{document} have gaps of type \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$(\mathrm{add}( \mathscr {M}), \omega )$\end{document}, and there are no (λ, ω)-gaps for \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\lambda < \mathrm{add}(\mathscr {M})$\end{document}, where \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathrm{add}(\mathscr {M})$\end{document} is the additivity number of the meager ideal. We also prove the existence of (ω1, ω1)-gaps in these structures. Finally we characterize the (...) cofinality of the meager ideal \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathrm{cof}(\ mathscr {M})$\end{document} using families of sets in \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathrm{Dense} (\mathbb {Q})/\mathbf {nwd}$\end{document}. (shrink)