On a superatomic Boolean algebra which is not generated by a well-founded sublattice

Abstract

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset$F \subseteq \omega ^\omega$ such that for everyg∈ω^ω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that$\aleph _1< 2^{\aleph _0 } = b$, and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ₀, ℵ₁, ℵ₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ₀, ℵ₀, λ, 1〉, andB is not well-generated, then λ≥b.