Possible pcf algebras

Abstract

There exists a family $\{B_{\alpha }{\}}_{\alpha <{\omega }_1}$ of sets of countable ordinals such that o $\max B_{\alpha }=\alpha$, o if $\alpha \in B_{\beta }$ then $B_{\alpha }\subseteq B_{\beta }$, o if $\lambda \le \alpha$ and $\lambda$ is a limit ordinal then $B_{\alpha }\cap \lambda$ is not in the ideal generated by the $B_{\beta }$, $\beta <\alpha$, and by the bounded subsets of $\lambda$, o 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.