Cardinal Sequences of length $ < \omega_2$ under GCH

István Juhász, Lajos Soukup and William Weiss

Let $ {\mathcal C}(\alpha)$ denote the class of all cardinal sequences of length $ \alpha$ associated with compact scattered spaces (or equivalently, superatomic boolean algebras). Also put

$\displaystyle {\mathcal C}_{\lambda}(\alpha)=\{s\in {\mathcal C}(\alpha): s(0)={\lambda} = \min[
s({\beta}) : \beta < {\alpha}]\}.$

We show that $ f\in {\mathcal C}(\alpha)$ iff for some natural number $ n$ there are infinite cardinals $ \lambda_0>\lambda_1>\dots>\lambda_{n-1}$ and ordinals $ {\alpha}_0,\dots {\alpha}_{n-1}$ such that $ {\alpha}={\alpha}_0+\cdots+{\alpha}_{n-1}$ and $ f=f_0\mathop{{}^{\frown}\makebox[-3pt]{}}\
f_1\mathop{{}^{\frown}\makebox[-3pt]{}}\cdots \ \mathop{{}^{\frown}\makebox[-3pt]{}}f_{n-1}$ where each $ f_i\in{\mathcal C}_{\lambda_i}(\alpha_i)$.

Under GCH we prove that if $ \alpha < \omega_2$ then

This yields a complete characterization of the classes $ {\mathcal C}(\alpha)$ for all $ \alpha < \omega_2$, under GCH.

Key words and phrases:

2000 Mathematics Subject Classification:

Downloading the paper