An essential base of the central types of the convex theory

Abstract

In this paper, we consider the model-theoretical properties of the essential base of the central types of convex theory. Also shows the connection between the center and Jonsson theory in permissible enrichment signatures. Moreover, the theories under consideration are hereditary. This article is divided into 2 sections: 1) an essential types and an essential base of central types (in this case, the concepts of an essential type and an essential base are defined using the Rudin-Keisler order on the set of central types of some hereditary Jonsson theory in the permissible enrichment); 2) the atomicity and the primeness of ϕ(x)-sets. In this paper, new concepts are introduced: the ϕ(x)-Jonsson set, the AP A-set, the AP A-existentially closed model, the ϕ(x)-convex theory, the ϕ(x)-transcendental theory, the AP A-transcendental theory. One of the ideas of this article refers to the fact that in the work of Mustafin T.G. it was noticed that any universal model of a quasi-transcendental theory with a strong base is saturated, but we generalized this result taking into account that: the concept of quasi-transcendence will be replaced by the ϕ(x)-transcendence, where ϕ(x) defines some Jonsson set; and the notion of a strong base is replaced by the notion of an essential base, but in a permissible enrichment of the hereditary Jonsson theory. The main result of our work shows that the number of fragments obtained under a closure of an algebraic or definable type does not exceed the number of homogeneous models of a some Jonsson theory, which is obtained as a result of a permissible enrichment of the hereditary Jonsson theory.