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## Cardinal sequences of length $< \omega_2$ under GCH

### Volume 189 / 2006

Fundamenta Mathematicae 189 (2006), 35-52 MSC: Primary 54A25, 54D30, 54G12; Secondary 06E05, 03E75. DOI: 10.4064/fm189-1-3

#### 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.

#### Authors

• István JuhászAlfréd Rényi Institute of Mathematics
V. Reáltanoda utca, 13–15
H-1053 Budapest, Hungary
e-mail
• Lajos SoukupAlfréd Rényi Institute of Mathematics
V. Reáltanoda utca, 13–15
H-1053 Budapest, Hungary
e-mail
• William WeissMathematics Department
University of Toronto