Generalized symmetric systems and thin-very tall compact scattered spaces

TL;DR

This paper constructs new examples of locally compact scattered spaces with specific height and thinness properties under GCH, solving a longstanding problem in the theory of compact scattered spaces and superatomic boolean algebras.

Contribution

It introduces a method to force the existence of ta very tall scattered spaces for any regular ta, extending previous constructions for ta = .

Findings

Existence of ta-thin very tall locally compact scattered spaces under GCH.

Development of a higher analogue of the Asperf3 and Bagaria poset ta for ta > .

Solution to a well-known problem in the theory of superatomic boolean algebras.

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higher analogue of the poset $\mathcal P_\omega$ originally introduced by Asper\'{o} and Bagaria in the context of an (unpublished) alternative consistency proof.