Cardinal sequences of length $< \omega_2$ under GCH
Volume 189 / 2006
Abstract
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 $$ {\cal C}_ {\lambda}(\alpha)=\{s\in \mathcal C(\alpha): s(0)={\lambda} = \min[ s({\beta}) : \beta < {\alpha}]\}. $$ 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 $f=f_0\kern-3pt\mathop{{}^{\frown}\kern-3pt} f_1\kern-3pt\mathop{{}^{\frown}\kern-3pt} \ldots \kern-3pt\mathop{{}^{\frown}\kern-3pt} f_{n-1}$ where each $f_i\in\mathcal C_{\lambda_i}(\alpha_i)$. Under GCH we prove that if $\alpha < \omega_2$ then
(i) $\mathcal C_{\omega}(\alpha)=\{s\in {}^{\alpha}\{{\omega},\omega_1\}: s(0)={\omega}\}$;
(ii) if $\lambda > \mathop{\rm cf} (\lambda)=\omega$, $$ {\cal C}_ {\lambda}(\alpha)=\{s\in {}^{\alpha}\{{\lambda},{\lambda}^+\}: s(0)={\lambda},\ s^{-1}\{\lambda\}\hbox{ is ${\omega}_1$-closed in ${\alpha}$} \}; $$ (iii) if $\mathop{\rm cf} (\lambda)=\omega_1$, $$ {\cal C}_ {\lambda}(\alpha)=\{s\in {}^{\alpha}\{{\lambda},{\lambda}^+\}: s(0)={\lambda},\, s^{-1}\{\lambda\}\hbox{ is ${\omega}$-closed and successor-closed in ${\alpha}$} \};$$ (iv) if $\mathop{\rm cf} (\lambda)>\omega_1$, $\mathcal C_\lambda (\alpha)= {}^\alpha\{\lambda\}$. This yields a complete characterization of the classes $\mathcal C(\alpha)$ for all $\alpha < \omega_2$, under GCH.
