Filtral Powers of Structures

Among the results of this paper are the following: 1. Every Boolean (u1tra)power is the union of an updirected elementary family of direct ultrapowers. 2. Under certain conditions, a finitely iterated Boolean ultrapower is isomorphic to a single Boolean ultrapower. 3. A w-bounded filtral power is an elementary substructure of a filtral power. 4. Let . X be an elementary class closed under updirected unions (e.g., if .X is an amalgamation class); then .X is closed under finite products if and only if .X is closed under reduced products if and only if .X is a Horn class. Introduction. Let M be a structure of finitary type, and let 9 be a complete Boolean algebra. Consider the set of functions defined on partitions in 9 with values in M . We extend each such function as follows: I f f is a map with domain a partition a , then for a E a and x 5 a , x # 0, we define f (x) = f (a). So each function defined on a partition is automatically also defined on the downwards closed set generated by the partition. To obtain the Bbolean power M [ 9 ] ,we shall impose an equivalence relation on the set of all such functions, by taking two functions two be equivalent provided that they agree on a.(hence any) refinement of their domains. M [9] is a structure of the same type as M (see Section 2 for more details). If 9 is a powerset algebra 9 ( X ) ,then 9 has a finest partition (namely all the atoms). It is clear that in this case the Boolean power M [ 9 ]is isomorphic to the direct power M X ,so that the notion of a Boolean power extends that of a direct power. It is not hard to see that every f E M [ 9 ]is equivalent to one that is injective on its domain: I f f has domain a, obtain a coarser domain y by putting a E y ifandonlyifa = C { x E a : f ( x ) =m)forsomemintherangeof f (i.e., y is obtained by taking joins of all a-elements on which f has the same value). We may then define a function g on y in the obvious way, so that g is injective, and f is equivalent to g . It is now easy to see that our definition of the Boolean power of M is equivalent to the standard definition, as found in, for example, Mansfield [12]. An extensive overview of the subject may be found in the papers by Burris [7, 81. Given any filter F on 9,we may define an equivalence relation on M [ 9 ]by taking f ,g to be equivalent provided C{ a E a : f (a) = g ( a )) E 9,where a is a common domain off and g . The structure obtained in this way, theJiltralpower Received May 8, 1996. The second author's work was supported by grants from the South African Council for Scientific and Industrial Research and the University of Cape Town Research Committee. @ 1998, Association for Symbolic Logic 0022-48 12/98/6304-0003/$2,6