A direct proof of the Feferman-Vaught theorem and other preservation theorems in products

Abstract

Here we give short and direct proofs of the Feferman-Vaught theorem and other preservation theorems in products of structures. In 1952, Mostowski [5] first showed the preservation of ≡ωω by direct powers of structures. Subsequently, in 1959, Feferman and Vaught [2] proved the preservation of ≡ωω by arbitrary direct products and also by reduced products with respect to cofinite filters. In 1962, Frayne, Morel and Scott [3] noticed that the results extend to arbitrary reduced products. In 1970, Barwise and Eklof [1] showed the preservation of ≡∞λ by products and in 1971 Malitz [4] showed the preservation of ≡κλ with κ strongly inaccessible (or ∞) by products. Below, we give short proofs of the above results. The ideas used here have initiated, in [7], [8], [9], [10], [11], the introduction of several new methods in the theory of products, which on the one hand give new, direct proofs of the known results in the area, including generalizations or strengthenings of some of those, and, on the other hand, give several new results as well in the theory of products and related areas.Below, L denotes a (first order) language, and by a structure we mean an L-structure. 0 and 1 denote the logically valid and false sentence, respectively. We may write ā ∈ A for ā ∈ An for some n. Also, the values 1 and 2 of a parameter l in the definitions below express a duality corresponding to disjunctive and conjunctive forms.