Filtral powers of structures

Among the results of this paper are the following: 1. Every Boolean (ultra)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 omega-bounded filtral power is an elementary substructure of a filtral power. 4. Let K be an elementary class closed under updirected unions (e.g., if K is an amalgamation class); then K is closed under finite products if and only if K is closed under reduced products if and only if K is a Horn class.