Possible pcf algebras

There exists a family $\{B_α\}_{α<ω_1}$ of sets of countable ordinals such that o $\max B_α=α$, o if $α\in B_β$ then $B_α\subseteq B_β$, o if $λ\leq α$ and $λ$ is a limit ordinal then $B_α\capλ$ is not in the ideal generated by the $B_β$, $β<α$, and by the bounded subsets of $λ$, o there is a partition $\{A_n\}_{n=0}^{\infty}$ of $ω_1$ such that for every $α$ and every $n,$ $B_α\cap A_n$ is finite.

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