The atomic definable subsets of semantic model In this paper some properties of small models, generally speaking, not necessarily complete theories and their relationship with each other were considered. Under small models we will understand some modifications of the concepts of countable atomic and prime models. These models were defined in the study of countable models of complete theories. Studies were conducted by analogy with the classic result of R. Vaught on countable-prime models of complete theories, but by other technical means. This work is oriented on the syntactic properties of special subsets of the semantic model of some Jonsson theory. A new concept was also introduced, as a model-theoretic rheostat , in order to obtain results related to the refinement of concept of atomicity within the framework of Jonsson theories. Thus, the main purpose of this article is to formally define this rheostat and to obtain on the basis of this concept the results having a relation to the refinement of the concept of atomicity in the frame of Jonsson theories.

The present paper concerns some properties of the so-called small models, generally speaking, not necessarily complete theories and their relationship with each other. In the well-known paper [1], R. Vaught have proved the fundamental theorem-criterion on the behavior of countable prime and atomic models for complete theories in countable language. The essence of this criterionis that in a complete theory any countable prime model is at the same time an atomic model of this theory. The result obtained in this paper is related to the classical problem of Vaught about countably prime models of complete theories but in more general formulation of the notion of countable atomicity. The main result of this paper is that it focuses on the syntactic properties on special subsets of a fragment of the semantic model the specific Jonsson theory. The concept of the so-called model-theoretic rheostat was also used to obtain results related to the refinement of the concept of atomicity in the framework of Jonsson's theories.