POSSIBLE PCF ALGEBRAS

There exists a family {B-alpha}(alpha<omega 1) of sets of countable or dinals such that: (1) max B-alpha = alpha, (2) if alpha is an element of B-beta then B-alpha subset of or equal to B-beta, (3) if lambda les s than or equal to alpha and lambda is a limit ordinal then B alpha bo olean AND lambda is not in the ideal generated by the B-beta, beta < a lpha, and by the bounded subsets of lambda, (4) there is a partition { A(n)}(infinity)(n=0) of omega(1) such that for every alpha and every n , B-alpha boolean AND A(n) is finite.