The structure of lattices of positive existential formulae of (Delta - PJ)-theories

Abstract

This article is related to one of the main branches of mathematical logic, model theory, and more precisely to what is called eastern model theory. This part of model theory is concerned with the study of incomplete inductive theories and more precisely Jonsson theories and some of their positive generalizations. It examines the model-theoretical properties of positive Jonsson theories. In particular, the lattice of special formulae is considered. In the study of complete theories one of the main methods is to use the properties of a topological space S-n(T). In the case of positive Jonsson theory, we can consider the lattice E-n(+)(T) of existential formulae, which is a sublattice of the Boolean algebra F-n(+)(T). The main aim of this article is to develop the basic concepts and methods of that part of model theory which will provide an opportunity for fruitful studies of Jonsson theories and some of its positive generalizations. Our technique is standard in the study of incomplete theories. The method consists of the translation of the elementary properties of the centre of a Jonsson theory into the theory itself.