A forcing construction of thin-tall Boolean algebras

It was proved by Juhász and Weiss that for every ordinal α with ${\displaystyle \{0<\alpha <{\omega }_2\}}$ there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that ${\displaystyle {\kappa }^{<\kappa }=\kappa }$ and α is an ordinal such that ${\displaystyle 0<\alpha <{\kappa }^{++}}$, then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height α and width κ. Furthermore, iterating this forcing through all ${\displaystyle \alpha <{\kappa }^{++}}$, we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic extension there is a superatomic Boolean algebra of height α and width κ for every ${\displaystyle \alpha <{\kappa }^{++}}$. Consistency for specific κ, like ${\displaystyle {\omega }_1}$, then follows as a corollary.