Constructions of Lindelöf scattered P-spaces

We construct locally Lindelöf scattered P-spaces (LLSP spaces, in short) with prescribed widths and heights under different set-theoretic assumptions. We prove that there is an LLSP space of width ω1 and height ω2 and that it is relatively consistent with ZFC that there is an LLSP space of width ω1 and height ω3. Also, we prove a stepping up theorem that, for every cardinal λ ≥ ω2, permits us to construct from an LLSP space of width ω1 and height λ satisfying certain additional properties an LLSP space of width ω1 and height α for every ordinal α < λ . Then, we obtain as consequences of the above results the following theorems: (1) For every ordinal α < ω3 there is an LLSP space of width ω1 and height α. (2) It is relatively consistent with ZFC that there is an LLSP space of width ω1 and height α for every ordinal α < ω4.