On the syntactic characterization of some model theoretic relations

Abstract

In this thesis we consider binary relations over the class of L - structures , for some fixed language L . Such a binary relation R, induces a binary relation R* between the class of theories in L; in the following natural way. If T1 and T2 are theories in L then T1R*T2 if ∃A,B A T1 , Bf T2 and ARB. We characterize syntactically those pairs of theories related by R* by introducing the concept of a notion of goodness for R. This consists of a set of ordered pairs of sentences in L, Delta, with the property that for theories T1 and T2 * T1R*T2 if for no empty1, empty2> epsilon Delta do we have T1 empty and T2empty. Provided is defined in a syntactically simple way, we find , by negating both sides of * and restricting the theories to sentences that the property * closely resembles an Interpolation Theorem for R. Actually, a notion of goodness is more complicated than this and our results are more general. In the established approaches to find Interpolation Theorems, the weak point has been in the understanding of "syntactically simple". We show, by considering certain relations which can be "described" by a theory in a particular language extending L, that a notion of goodness can often be found immediately from such a theory. Indeed we find a model theoretic condition on R for which this is possible. It turns out to be a "union of chains" condition. Using this approach we obtain many Interpolation Theorems by analysing the structure of the theories used to "describe" R. In particular the methods are used to prove a version of Feferman's Interpolation Theorem in a many-sorted language. We give a characterization of those theories with the Amalgamation Property and the Strong Amalgamation Property. We conclude with a solution of an open problem of G. Gratzer.