A Forcing Construction for Thin-Very Tall Boolean Algebras

Juan Carlos Martínez

A Boolean algebra B is superatomic, if every subalgebra of B is atomic. It is known that the Cantor-Bendixson process for topological spaces can be transferred to the context of Boolean algebras. If B is a Boolean algebra, we define for every ordinal α the ideal I_{α} as follows: I_{0} = {0}; if α = β + 1, I_{α} = the ideal generated by I_{β} ∪ { b ∈ B : b/I_{β} is an atom in B/I_{β} }; and if α is a limit, I_{α} = ⋃{ I_{β} : β < α }. Then, B is superatomic iff there is an ordinal α such that B = I_{α}.

We define the height of a superatomic Boolean algebra B by ht(B) = the least ordinal α such that B/I_{α} is finite (which means B = I_{α+1}). For every α < ht(B), we denote by wd_{α}(B) the cardinality of the set of atoms of B/I_{α}, and then we define the width of B by wd(B) = sup{ wd_{α}(B) : α < ht(B) }.

It was shown by Juhász and Weiss that for every ordinal α < ω_{2}, there is a superatomic Boolean algebra of width ω and height α. Also, Baumgartner and Shelah proved that Con (ZFC) → Con (ZFC + “there is a superatomic Boolean algebra of width ω and height ω_{2}”). Then, we will show that we can improve Baumgartner-Shelah’s theorem, showing that Con (ZFC) → Con (ZFC + “for every α < ω_{3} there is a superatomic Boolean algebra of width ω and height α”).

Address: Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

E-mail: martinez@mat.ub.es