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 $\kappa \ge \omega$, 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analogue of the poset ${\mathcal{P}}_{\omega }$ originally introduced by Asperó and Bagaria in the context of an (unpublished) alternative consistency proof.

14 pages