Superatomic Boolean algebras constructed from strongly unbounded functions

TL;DR

This paper demonstrates the consistency of constructing superatomic Boolean algebras with specific height and level cardinalities using strongly unbounded functions, expanding the known possibilities for their cardinal sequences.

Contribution

It introduces a method to build superatomic Boolean algebras with prescribed cardinal sequences via forcing and strongly unbounded functions, under certain set-theoretic assumptions.

Findings

Existence of superatomic Boolean algebras with specified height and level sizes.

Cardinal sequences like _ and _ concatenations are realizable.

The construction relies on set-theoretic forcing and 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 $\kappa^{+++} \leq \lambda$, $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}= \kappa^+$, and $\eta$ is an ordinal with $\kappa^+\leq \eta <\kappa^{++}$ 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, $\<{\omega}\>_{{\omega}_1}\concatenation \<{\omega}_3\>$ and $\<{\omega}_1\>_{{\omega}_2}\concatenation \<{\omega}_4\>$ can be cardinal sequences of superatomic Boolean algebras.