Two Remarks on Partitions of ω with Finite Blocks

We prove that all algebras P(w)/I_R, where the I_R-'s are ideals generated by partitions of W into finite and arbitrary large elements, are isomorphic and homogeneous. Moreover, we show that the smallest size of a tower of such partitions with respect to the eventually-refining preordering is equal to the smallest size of a tower on ω.