Some applications of infinitely long formulas

Introduction. This paper is a sequel to our paper [3]. In that paper we introduced the notion of a finite approximation to an infinitely long formula, in a language L with infinitely long expressions of the type considered by Henkin in [2]. The results of the paper [3] show relationships between the models of an infinitely long sentence and the models of its finite approximations. In the present paper we shall apply the main result of [3] to prove a number of theorems about ordinary finitely long sentences; these theorems are of a type which one might call "preservation theorems". The following two known results are typical preservation theorems: A sentence qp is preserved under substructures if and only if q is (logically) equivalent to some universal sentence (Los [7] and Tarski [11]); q9 is preserved under homomorphic images if and only if p is equivalent to some positive sentence (Lyndon [8]). An expository account of preservation theorems may be found in Lyndon [9]. We shall use freely all of the notation introduced in [3], and shall not attempt to make this paper selfcontained. However, the reader should have no difficulty following this paper after he has read [3]. In ? 1 we cover some preliminary topics. In ? 2 and ? 3 we shall illustrate our method in familiar situations by proving anew the two preservation theorems mentioned above. We shall then obtain four new preservation theorems in ?? 4, 5, and 6, involving respectively direct powers and roots, strong homomorphisms and retracts, and direct factors. The result in ? 6 was stated in the abstract [4], while the results in ? 5 were stated in the abstract [5]. ? 1. Preliminaries. We let m be an arbitrary but fixed infinite cardinal. We consider the language L' which is exactly like the language L of [3], depending on the cardinal m, with the following two exceptions. First, L' has only the singly binary predicate symbol P, instead of an arbitrary list of at most m finitary predicate symbols. Second, L' has a doubly indexed, rather than singly indexed, sequence of variables va, a, f < m+. We let X = m+ x m+. These two changes are made only to