Abstract
2. Preliminaries. Throughout the paper, a and /3 denote infinite cardinal numbers. We write aV/3 for max [a, 0]. The symbol oo is used as if it were a cardinal exceeding all other cardinal numbers. For any a, define a+ to be the least cardinal exceeding a. The terms, a-ideal and a-field are used in the usual way, that is, the operations in these structures enjoy closure up to and including the power a. In particular, the term complete field means the field of all subsets of some set. The notions of a-homomorphism, a-isomorphism and asubalgebra will have special meanings which are explained in 3.1 below. Lattice operations of join, meet, inclusion and complement are designated by V, A, ^ and (') respectively. Corresponding set operations are represented by rounded symbols: VJ, T\, C and (c). The symbols 0 and 0 will always stand for the zero of a Boolean algebra, while u and u denote the unit element. The empty set is designated by 0. If S and T are sets, T — S is the set of elements in T, but not in S. If h is a mapping of 5 into T, and if XQS, YCZT, we denote by h(X) the set {h(x)\xEX} and by h~1(Y) the set {xEX\h(x)EY}. The symbol \S designates the cardinality of the set S.
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