About central types and the cosemanticness of the ∆-PM fragment of the Jonsson set

This article is concerned with the enrichment of the signature. In own time, when studying the stability of the theory and the concept of an elementary pair of models, Mustafin T.G. had noticed that these things are related to each other and he introduced the concept T ∗-stability [1]. In fact, some enrichment of the signature is considered. Generally speaking, the theories obtained in the extended language are incomplete, therefore, the number of such completions of these theories is sought. This number also determines stability in the sense of T ∗-stability. It was noted by E.A.Palyutin in [2] that the concept of T ∗-stability is not invariant with respect to definability of type. But we know that in the classical sense of S.Sellach the stability of the theory is invariant with respect to the definability of type. Therefore Palyutin E.A. had introduced the concept E∗-stability, which preserved the definability of type. Author of this article [3] considered this formulation of the problem for the Jonsson theories. We call it in the class of Jonsson theories or in positive Jonsson theories (∆-PJ , ∆-PM , ∆-PR) enrichment of the signature is admissible if the stability was obtained in the considering case is invariant with respect to the definability of type. In this article, all considering enrichments are admissible. Let the enrichment be Γ = {P} ∪ {c}, where P is unary predicate symbol with new constant symbol. In connection with admissible enrichments one of the authors of this paper introduced the notion of the central type. Many theorems which obtained before the enrichment of the signature are translated in the language of central types. In this article we will consider similarly questions for central types of positive generalizations of Jonsson fragments.