Higher degrees of distributivity and completeness in Boolean algebras

Abstract

Introduction(%). Let ac and A be cardinals. A Boolean algebra A is called ai-complete if the sum of every set of at most a elements of A exists in A. An a-complete ideal in a Boolean algebra is analogously defined. The class of all ai-complete Boolean algebras is a subclass of the class of all Boolean algebras, and this subclass becomes smaller as a increases. A Boolean algebra A is called (a, ,3)-distributive if the product of the sums of at most ae sets X, each consisting of at most A elements of A, is equal to the sum of all possible products, each of which contains precisely one factor from each X, provided that the sums and products in question exist. In this paper we study a-complete and (a, ,B)-distributive Boolean algebras, restricting ourselves mainly to the case in which ai and A are infinite. So far only the case in which both a and ,3 are the smallest infinite cardinal has been systematically studied. However, a number of results are known which apply to special a-complete Boolean algebras, a-complete fields of sets. We shall show, for example, that every (a, 2)-distributive Boolean algebra A is (a, a)-distributive, and, moreover, if A is a-complete, then it is also (a, 2(a))distributive. Also, if a Boolean algebra A is 2(a)-complete, then every acomplete prime ideal in A is 2(a)-complete. This paper is divided into four sections. The first concerns terminology and symbolism and contains the definition and elementary facts concerning a-complete Boolean algebras. Section two explores (a, 3)-distributivity, while section three primarily contains some results which permit one to conclude that an ideal in a Boolean algebra is a-complete. Some conditions under which a factor algebra is a-complete are established in ?4. Miscellaneous applications to measure theory are included. 1. Terminology and symbolism. We use ordinary set-theoretic notions and symbolism. In particular, C, C (or D), 'U, n and 0 will denote the relations of membership and inclusion, the operations of union and intersec-