Comparing Classes of Finite Sums

Abstract

The notion of Turing computable embedding [4] provides an eective way to compare classes of countable structures, reducing the classication problem for one class to that for the other. Most of the known results on non-existence of Turing computable embeddings reect dierences in the complexity of the sentences needed to distinguish among non-isomorphic members of the two classes. Here we give some examples of further distinctions that we can make using Turing computable embeddings. The classes that we consider consist of sum structures. We consider cardinal sums of n structures, in which the components are named by predicates, and sums given by an equivalence relation, where the components are not named. We also consider direct sums of certain groups. The results are based on model-theoretic considerations related to Morley degree. The proofs of non-embeddability involve index set calculations.