Cardinal Sequences of Lindel\"of scattered P-spaces

TL;DR

This paper explores the structure of cardinal sequences in locally Lindelöf, scattered P-spaces, introducing new construction methods, limitations, and conditions for possible sequences.

Contribution

It provides a novel construction approach for LLSP spaces and establishes both necessary and sufficient conditions for their cardinal sequences.

Findings

Constructed LLSP spaces from cone systems and partial orders.

Identified limitations on cardinal sequences of LLSP spaces.

Established conditions for sequences to be realized as cardinal sequences.

Abstract

We continue our investigation of cardinal sequences associated with locally Lindelof, scattered, Hausdorff P-spaces (abbreviated as LLSP spaces). We outline a method for constructing LLSP spaces from cone systems and partial orders with specific properties. Additionally, we establish limitations on the cardinal sequences of LLSP spaces. Finally, we present both a necessary condition and a distinct sufficient condition for a sequence $\langle \kappa_\alpha: \alpha < \omega_2 \rangle$ to be the cardinal sequence of an LLSP space.