Model-theoretic properties of the #-companion of a Jonsson set

The work is devoted to the model theory. The subject of this article is connected with the study of incomplete inductive theories. In particular, model-theoretic properties of Jonsson theories are considered, which are subclasses of inductive theories. This paper considers a fragment of a certain Jonsson subset of a semantic model of a fixed Jonsson theory, and, as a class of models, all models of the given fragment are considered, namely, the paper considers the model-theoretic properties of countable and uncountable categoricity and the properties of elimination of quantifiers of the given fragment's #-companions. Also the properties of #-companions of the fragment's existential formulas are investigated.