A forcing construction of thin-tall Boolean algebras

It was proved by Juhász and Weiss that for every ordinal α with 0 < α < ω2 there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that κ = κ and α is an ordinal such that 0 < α < κ++, 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 α < κ++, 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 α < κ++. Consistency for specific κ, like ω1, then follows as a corollary. 0. Introduction. A superatomic Boolean algebra is a Boolean algebra in which every subalgebra is atomic. It is a well-known fact that a Boolean algebra B is superatomic iff its Stone space S(B) is scattered. For every ordinal α, the α-derivative of S(B) is defined by induction on α as follows: S(B)0 = S(B); if α = β + 1, then S(B) is the set of accumulation points of S(B) ; and if α is a limit, then S(B) = ⋂{S(B)β : β < α}. Then S(B) is scattered iff S(B) = ∅ for some α. This process can be transferred to the Boolean algebra B, yielding an increasing sequence of ideals Iα, which are defined by transfinite induction as follows: we put I0 = {0}; if α = β + 1, then Iα = the ideal generated by Iβ ∪ {b ∈ B : b/Iβ is an atom in B/Iβ}; and if α is a limit, then Iα = {Iβ : β < α}. Then B is superatomic iff there is an ordinal α such that B = Iα. We define the height of a superatomic Boolean algebra B by ht(B) = the least ordinal α such that B/Iα is finite (which means B = Iα+1). For every α < ht(B), we denote by wdα(B) the cardinality of the set of atoms of B/Iα, and we define the width of B by wd(B) = sup{wdα(B) : α < ht(B)}. If κ is an infinite cardinal and η 6= 0 is an ordinal, we say that a Boolean algebra B is a (κ, η)-Boolean algebra if B is superatomic, wd(B) = κ and 1991 Mathematics Subject Classification: 03E35, 06E99,54G12. The preparation of this paper was supported by DGICYT Grant PB94-0854.