Extensions of Free Boolean Algebras

In a series of papers [2, 3, 4, 5], B. Efimov has given sufficient conditions for a Boolean algebra to have a free subalgebra of given cardinality. For example, he has shown that if an infinite Boolean algebra A has cardinality greater than exp exp exp v, but the supremum of the cardinalities of subsets of disjoint elements in A is not greater than v, then A has a free subalgebra with (exp v) independent generators, where for any cardinal v, v is its successor. His work has all been done in a topological setting. On the other hand, one of several characterizations of the class of superatomic Boolean algebras is that this is the class of all Boolean algebras which do not have a countable, infinite, free Boolean algebra as a subalgebra (see G. Day [1]). In this paper, we generalize the concept of the superatomic radical, for the purpose of obtaining further results about the class of all Boolean algebras which do not contain a given free Boolean algebra as a subalgebra. Equivalent formulations of this class are given, and in an attempt to determine exactly which algebras belong to this class, the closure properties of the class under subalgebras, homomorphic images, sums, products, and weak products are examined. Many definitive results concerning these closure properties are produced.