Model-theoretic properties of J-superstable Jonsson theories in classes defined by cosemanticness

This article deals with the problems of model-theoretic characterization of J-superstable Jonsson theories. The characteristic features of such theories are analyzed in terms of J-stability, J-P-superstability, and J-nonmultidimensionality. The need to generalize classical stability notions to the framework of Jonsson theories is identified and justified. The concepts of J-stationary and J-orthogonal types are introduced, and their role in describing the dimensional structure of existentially closed models is examined. It is shown that every JFλα-saturated model embeds as a submodel of the semantic model of a Jonsson theory T. Based on the notion of J-stationarity, a theory of independence for existential types is developed, and the notions of J-basis and J-dimension are defined. The equivalence between J-nonmultidimensionality, J-P-superstability, and J-P-stability is established, providing precise criteria for the model-theoretic classification of Jonsson theories. The results contribute to the refinement of model-theoretic tools for analyzing stability and dimensionality within the framework of Jonsson theories and place these findings in the broader context of modern classification theory, highlighting Jonsson theories as a natural generalization of elementary theories. These results clarify the interaction between saturation, independence, and dimensionality in Jonsson theories and provide a unified framework for further developments in their model-theoretic classification.