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

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset F ⊆ ω^ω such that for every g ∈ ω^ω there is f ∈ F such that {n: g(n) ≤ f(n)} is infinite. A Boolean algebra B is well-generated, if it has a well-founded sublattice L such that L generates B. We show that it is consistent with ZFC that א₁ < 2^א0 = b, and there is a Boolean algebra B such that B is not well-generated, and B is superatomic with cardinal sequence 〈א₀, א₁, א₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebra B is 〈א₀, א₀, λ, 1〉, and B is not well-generated, then λ ≥ b.