The property of independence for Jonsson sets

Abstract

The studies carried out in this article are connected with the description of model - theoretic properties of some, generally speaking, incomplete classes of theories that make a subclass of inductive theories. These theories are well studied both in algebra and in the theory of models. They are called Jonsson’s theories. To study these theories there is introduced a new research approach, namely: on the submultitudes of a semantic model of Jonsson’s theory there are separated special multitudes that are, firstly, realizations of some existential formula, secondly, the closing of the set gives us the basic set of some existentially closed submodel of the semantic model. Besides, there is developed a technique of studying the central orbital types. It is well known that the perfect Jonsson theory enough comfortable for model - theoretic researches. Practically, in the perfect case, we can say that with the help of semantic method, we can give a specific description of these objects (Jonsson theory and class its existentially closed models). In this article we will give the notion of forking for fragment of fixing Jonsson theory. The nonforking extensions will be the «Mfree» ones. Also we considered for the notion of independence many desirable properties like monotonicity, transitivity, finite basis and symmetry.