Model theory for universal classes with the amalgamation property: A study in the foundations of model theory and algebra

Abstract

In this paper a complete model theory is constructed for classes of algebras and relational structures satisfying certain natural algebraic conditions. The paper a detailed exposition of the model theory of universal classes with the amalgamation property and applies this theory to the study of the foundations of various classical algebraic theories. As well the interrelationship of this new theory with the model theory of complete full theories is fully explored and the similarities and differences of the structure of these model theories is carefully delineated. We conclude that the model theory of universal classes with the amalgamation property has a structure closely related to the classical theory. Indeed many of the concepts introduced in the course of the paper are strict generalizations of classical notions. As wells it is shown that the full first order languages associated with the model theory of complete full theories have properties similar to the properties of the open languages associated with the model theory of universal classes with AP. Besides the construction of a new model theory the paper is intended to explore the consequences of the assumption of the amalgamation property, an algebraic mapping property which arises naturally in both modedl theory and algebra. In this regard it is shown that the structural and linguistic consequences of the assumption of this property are intimately connected. This provides us with new insights into the foundations of model theory. Finally, the construction of the model theory of universal classes with the amalgamation property establishes a very general framework and tool for the study of many algebraic and model theoretic notions. In the course of the paper we prove structure theorems, splitting theorems, and results concerned with the properties of extensions of models. Many of these results are new even in the classical case. In particular we developed a detailed local structure theory and prove that this theory extends the basic structure theory of the classical theory of affine varieties in algebraic geometry. Our results in this area provide a firm foundation for the study of a new class of geometric objects as well as resolving the representation problem for constructible subsets of affine varieties. We include many conjectures and open problems in the text of the paper.