Abstract

Among the results of this paper are the following: 1. Every Boolean 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 $\mathscr{K}$ be an elementary class closed under updirected unions ; then $\mathscr{K}$ is closed under finite products if and only if $\mathscr{K}$ is closed under reduced products if and only if $\mathscr{K}$ is a Horn class.