Superatomic Boolean algebras constructed from strongly unbounded functions

Abstract

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf(η) = κ+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal B$\end{document} such that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{ht}(\mathcal B) = \eta + 1$\end{document}, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{wd}_{\alpha }(\mathcal B) = \kappa$\end{document} for every α < η and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{wd}_{\eta }(\mathcal B) = \lambda$\end{document}(i.e., there is a locally compact scattered space with cardinal sequence 〈κ〉η⌢︁〈λ〉). Especially, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\langle {\omega }\rangle _{{\omega }_1}{}^{\smallfrown } \langle {\omega }_3\rangle$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\langle {\omega }_1\rangle _{{\omega }_2}{}^{\smallfrown } \langle {\omega }_4\rangle$\end{document} can be cardinal sequences of superatomic Boolean algebras.