The atomic definable subsets of semantic model

Abstract

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.

Closure of special atomic subsets of semantic model

“…Proof Suppose A is (∇ 1 , ∇ 2 ) − cl − ∆ − nice − atomic. Then, from the same proof of an isomorphism of the corresponding countable models by Theorem 4 [5] and Theorem 3 (the proof of which can be found in the work of "Core Jonsson theories"in this volume) it follows that A is (∇ 1 , ∇ 2 ) − cl − ∆ − nice. The rest follows from the perfection of the fragment F and the model completeness of F * .…”

“…Since A is in a particularly algebraically prime and existentially closed model of F , we know that F has (∇ 1 , ∇ 2 ) − cl − ∆ − nice− atomic model. Therefore by Theorem 4 [5] every existential formula ψ(x) consistent with F is implied by an existential formula ϕ(x) complete for (∆)-formulas. By (R 0 ) is, in fact, complete for existential formulas.…”

“…In this article, we will focus our attention on the study of special models for certain types of Johnson theory within the framework of the above topics [6][7][8]. To have an idea of previous works concerning the behavior of small models in Johnson's theories, the following sources can be used: [4,5]. One of the central ideas that allow us to compare the concepts of atomicity in the sense of [1] and in the sense of [2] is the idea concept of rheostat [4,5].…”

“…To have an idea of previous works concerning the behavior of small models in Johnson's theories, the following sources can be used: [4,5]. One of the central ideas that allow us to compare the concepts of atomicity in the sense of [1] and in the sense of [2] is the idea concept of rheostat [4,5]. It is clear that the larger the Johnson set, the closer the model considered to atomicity from [2] and, conversely, the smaller it is, the closer to the notion of atomicity from [1].…”

“…2) for all a 0 , ..., a n−1 ∈ A, b 0 , ..., b n−1 ∈ A , if (M, a 0 , ..., a n−1 ) ⇒ ∃ (M , b 0 , ..., b n−1 ), then ∀a n ∈ A, ∃b n ∈ A such that (M, a 0 , ..., a n ) Principle of rheostat . [5] Let two countable models A 1 , A 2 of some Jonsson theory T be given. Moreover, A 1 is an atomic model in the sense of [1], and X is (∇ 1 , ∇ 2 ) − cl -algebraically prime set of theory T and cl(X) = A 2 .…”