Generalized symmetric systems and thin-very tall compact scattered spaces

Abstract

We solve a well--known problem in the theory of compact scattered spaces and superatomic boolean algebras by showing that, under GCH and for each regular cardinal $\kappa \geq \omega$, there is a poset $\mathcal P_\kappa$ preserving all cardinals and forcing the existence of a $\kappa$--thin very tall locally compact scattered space. For $\kappa > \omega$, we conceive the poset $\mathcal P_\kappa$ as a higher analogue of the poset $\mathcal P_\omega$ originally introduced by Asper\'{o} and Bagaria in the context of an (unpublished) alternative consistency proof.