Possible pcf algebras

by Jech and Shelah. [JeSh:476]
J Symbolic Logic, 1996
There exists a family {B_{alpha}}_{alpha < omega_1} of sets of countable ordinals such that 1) max B_{alpha}= alpha, 2) if alpha in B_{beta} then B_{alpha} subseteq B_{beta}, 3) if lambda <= alpha and lambda is a limit ordinal then B_{alpha} cap lambda is not in the ideal generated by the B_{beta}, 4) beta < alpha, and by the bounded subsets of lambda, 5) there is a partition {A_n}_{n=0}^{infty} of omega_1 such that for every alpha and every n, B_{alpha} cap A_n is finite.

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