Application of the Cantor-Bendixon Construction to the Problem of Freely Generated Filters

In [1] and [2] we have discussed the problem of existence of an independent set of generators for a countably generated filter in an atomless Boolean algebra and for any filter in a free Boolean algebra. In particular, we have proved that if F is a filter in a free Boolean algebra then F is freely generated provided the minimal cardinality of the set of generators of F is not a singular cardinal with a countable cofinality. Moreover, it has been proved in [1] that every countably generated filter in an atomless Boolean algebra is freely generated in this algebra. Now, we would like to consider the problem of freely generated filters in Boolean algebras containing atoms. It is easy to observe (see [1]) that Received September 24, 2001 1991 AMS Subject Classification 03G05, 03E99