Direct product decompositions of structures and theories (Q1918968)

The presented results are related to investigations of Feferman-Vaught (1959), Galvin (1970), Keisler (1965), and Makkai (1965), and concern new methods in the theory of decompositions of arbitrary structures. First, the author gives an introduction to the paper and refers to results described in several other papers submitted by himself, especially an extension of König's Infinity Lemma. Using some basic definitions and results in the logic of product structures, the operations of product, quotient, roots, and projections for sentences are defined and calculated. Several other properties concerning finiteness and definability are studied. By a product, factor, etc. sentence one has to understand a sentence preserved by direct products, direct factors, etc., respectively. Then the logical complexity of product theoretic operations are characterized. In a further section the author gives some criteria for several aspects of decomposability such as direct factors of a structure, decomposability and indecomposability of a structure in general, and decomposability of a theory. Using product nets of a theory \(T\) the decomposition spectrum \(\text{SP} (T,S)\) with respect to a theory \(S\) is calculated. Further the author proves some finiteness conditions for the decomposability of theories. Finally he investigates certain model-theoretic properties of the classes \(\prod_{j\in\mu} K_j\), \(K^\mu\), \(K^\infty\), \(K^{\underline\mu}\), \(K^{\underline \infty}\), \(K^{\geq\mu}\), \({K\over L}\), \(\root\mu \of {K}\) (defined in a natural way from classes of structures \(K_j (j\in\mu)\), \(K\), and \(L)\).