Possible pcf algebras

There exists a family {B_α}_α<ω1 of sets of countable ordinals such that: (1) max Bα = α, (2) if α ∈ B_β then B_α ⊆ B_β, (3) if λ ≤ α and λ is a limit ordinal then B_α ∩ λ is not in the ideal generated by the B_β, ß < α, and by the bounded subsets of λ, (4) there is a partition {A_n}^∞_n=0 of ω₁ such that for every α and every n, B_α, ∩ A_n is finite.