Boolean algebras with no rigid or homogeneous factors

Abstract

A simple construction of Boolean algebras with no rigid or homogeneous factors is described. It is shown that for every uncountable cardinal κ \kappa there are 2 κ {2^\kappa } isomorphism types of Boolean algebras of power κ \kappa with no rigid or homogeneous factors. A similar result is obtained for complete Boolean algebras for certain regular cardinals. It is shown that every Boolean algebra can be completely embedded in a complete Boolean algebra with no rigid or homogeneous factors in such a way that the automorphism group of the smaller algebra is a subgroup of the automorphism group of the larger algebra. It turns out that the cardinalities of antichains in both algebras are the same. It is also shown that every κ \kappa -distributive complete Boolean algebra can be completely embedded in a κ \kappa -distributive complete Boolean algebra with no rigid or homogeneous factors.