Superatomic Boolean algebras constructed from strongly unbounded functions

Abstract

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $\kappa ,\lambda$ are infinite cardinals such that ${\kappa }^{+++}\le \lambda$, ${\kappa }^{<\kappa }=\kappa$ and $2^{\kappa }={\kappa }^{+}$, and $\eta$ is an ordinal with ${\kappa }^{+}\le \eta <{\kappa }^{++}$ and $cf(\eta )={\kappa }^{+}$. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra $B$ such that - $ht(B)=\eta +1$, - the cardinality of the $\alpha$th level of $B$ is $\kappa$ for every $\alpha <\eta$, - and the cardinality of the $\eta$th level of $B$ is $\lambda$ Especially, $\<ω\>_{ω_1}\concatenation \<ω_3\>$ and $\<ω_1\>_{ω_2}\concatenation \<ω_4\>$ can be cardinal sequences of superatomic Boolean algebras.

13 pages