On uncountable cardinal sequences for superatomic Boolean algebras (Q1902339)

The countable sequences of cardinals which arise as cardinal sequences of superatomic Boolean algebras were characterized by La Grange on the basis of ZFC set theory. However, no similar characterization is available for uncountable cardinal sequences. In this paper we prove the following two consistency results: (1) If \(\theta = \langle \kappa_\alpha : \alpha < \omega_1\rangle\) is a sequence of infinite cardinals, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebra \(B\) such that \(\theta\) is the cardinal sequences of \(B\). (2) If \(\kappa\) is an uncountable cardinal such that \(\kappa^{<\kappa}=\kappa\) and \(\theta=\langle\kappa_\alpha:\alpha<\kappa^+\rangle\) is a cardinal sequence such that \(\kappa_\alpha\geq\kappa\) for every \(\alpha<\kappa^+\) and \(\kappa_\alpha=\kappa\) for every \(\alpha<\kappa^+\) such that \(\text{cf}(\alpha)<\kappa\), then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebra \(B\) such that \(\theta\) is the cardinal sequence of \(B\).