Properties of central types of positive Jonsson theory. Yeshkeyev - - PowerPoint PPT Presentation

Properties of central types of positive Jonsson theory.

Yeshkeyev Aibat Rafhatuly

Karaganda State University, The Institution of Applied Mathematics Karagandy, Kazakhstan

June 19-22, 2014 AAA88 Workshop on General Algebra

Warsaw University of Technology Faculty of Mathematics and Information Science Warsaw, Poland

Yeshkeyev А.Р. aibat.kz@gmail.com

Outline

Positiveness The enrichment of the signature The question of A.D.Taymanov The criterium of perfectness Stability of the central type and forcing companion Small models Categoricity

Yeshkeyev А.Р. aibat.kz@gmail.com

Agreements about positiveness and ∆

Let L is the language of the first order. At is a set of atomic formulas of this language. B+(At) is containing all the atomic formulas, and closed under positive Boolean combination and for sub-formulas and substitution

• f variables.

L+ = Q(B+(At) is the set of formulas in prenex normal type

• btained by application of quantifiers (∀ and ∃) to B+(At).

B(L+) is any Boolean combination of formulas from L+ ∆ ⊆ B(L+)

Yeshkeyev А.Р. aibat.kz@gmail.com

∆-Homomorphism

Let M and N are the structure of language, ∆ ⊆ B(L+). The map h : M → N ∆-homomorphism (symbolically h : M ↔∆ N, if for any ϕ(¯ x) ∈ ∆, ∀¯ a ∈ M such that M | = ϕ(¯ a), we have that N | = ϕ(h(¯ a))). The model M is said to begin in N and we say that M continues to N, with h(M) is a continuation of M. If the map h is injective, we say that h immersion M into N (symbolically h : M ← →∆ N). In the following we will use the terms ∆-continuation and ∆-immersion continued the dive.

Yeshkeyev А.Р. aibat.kz@gmail.com

∆-Joint Embedding Property and ∆-Amalgamation Property

Definition 1. The theory T admits ∆ − JEP, if for any A, B ∈ ModT there are exist C ∈ ModT and ∆-homomorphism’s h1 : A →∆ C, h2 : B →∆ C. Definition 2. The theory T admits ∆ − AP, if for any A, B, C ∈ ModT with h1 : A →∆ C, g1 : A →∆ B, where h1, g1 are ∆-homomorphism’s, there are exist D ∈ Modt and h2 : C →∆ D, g2 : B →∆ D where h2, g2 are ∆-homomorphism’s such that h2 ◦ h1 = g2 ◦ g1.

Yeshkeyev А.Р. aibat.kz@gmail.com

∆ − M-theory

Definition 3. Theory T called ∆-positive mustafinien (∆ − M)-theory, if the morphisms considered only immersion and following conditions are true: (1) theory T has infinite models, (2) theory Tis Π+

n+2-axiomatizable,

(3) theory T admits ∆-JEP, (4) theory T admits ∆-AP. Easy to see that ∆-Jonsson theory is a special case ∆-Mustafinien theory.

Yeshkeyev А.Р. aibat.kz@gmail.com

Positively existentially closeness

Definition 4. The model A of the theory T is ∆-positively existentially closed, if for any ∆-homomorphism’s h : A →∆ B and any ¯ a ∈ A and ϕ(¯ x, ¯ y) ∈ ∆, B | = ∃¯ yϕ(h(¯ a), ¯ y) ⇒ A | = ∃¯ yϕ(¯ a, ¯ y).

Yeshkeyev А.Р. aibat.kz@gmail.com

New signature

The enrichment of the signature Let T is arbitrary ∆ − M-theory in the language of the signature σ. Let C - semantic model of the theory T. A ⊆ C. Let σΓ(A) = σ {ca| a ∈ A} Γ, where Γ = {g} {c} {P}. Consider the following theory T PgM

Γ

(A) = ThΠ+

α+2(C, a)a∈A

{g(a) = a | a ∈ A} g(c) Tg {P(c)} {”P ⊆ ”}, where Tg- expresses the fact that for any model (M, gM) | = Tg holds: (1) gM-automorphism of M; (2) there is {m ∈ M | gM(m) = m} a universe of existentially closed submodel M, for any model M of signature σ.

Yeshkeyev А.Р. aibat.kz@gmail.com

New signature

The enrichment of the signature Predicate P, we write an expression {”P ⊆ ”}, that inherently has an infinite set of sentences, which says that the interpretation of characters have positively existentially closed submodel in the

• signature. By incomplete we do not write the exact relationships

between the elements Γ = {g} {c} {P}, but it is assumed that they are consistent in terms of the theory T PgM(a).

Yeshkeyev А.Р. aibat.kz@gmail.com

Central type

Central type Consider all completions of the center T ∗ of the theory T in the new signature σΓ, where Γ = {c}. By virtue ∆ − M-ness of the theory T ∗, there is its center, and we denote it as T c. When restricted T c up the signature σ, the theory T c becomes a complete type. This type we call as the central type the theory T.

Yeshkeyev А.Р. aibat.kz@gmail.com

The question of A.D.Taymanov

Formulate the question A.D.Taymanov, given that the problem has been defined for complete theories. Namely, in the study of the properties of models of complete theories first-order are useful information on Boolean algebras (algebras Lindenbaum-Tarski) Fn(T), n ∈ ω, theory Т . In connection with these Boolean algebras Fn(T), n ∈ ω, is well-known question A.D. Taimanov : (*) What properties must have Boolean algebras Bn,n ∈ ω that there exists a complete theory T, so that Bn was isomorphic Fn(T), n ∈ ω?

Yeshkeyev А.Р. aibat.kz@gmail.com

The question of A.D.Taymanov

T.G.Mustafin were given answers to particular cases of this issue. He obtained the following results: Theorem 1. For any Boolean algebra B there exists a complete theory T, that: (a) B ∼ = F1(T), (b) if B the finite, then T is categorical in the countable power, (c) if the Stone space of the algebra B is countable, then T is totally transcendental. Theorem 2. In order to finite Boolean algebras B1, B2 there is a categorical in a countable power theory T, such that F1(T) ∼ = B1, F2(T) ∼ = B2 is necessary and sufficient that the number of atoms of B2 was greater than the square of the number of atoms B1.

Yeshkeyev А.Р. aibat.kz@gmail.com

The question of A.D.Taymanov

The natural generalization of the question of A.D.Taymanov would be to consider this problem in the framework of non complete theories, in particular in Jonsson theories and in the frame of their positive generalizations. It is well known that working with Jonsson theories in some cases we are able to limit yourself to existential formulas and existentially closed models considered Jonsson theory. In this case, instead of the Lindenbaum-Tarski algebras Fn(T), n ∈ ω, should be considered lattice existential formulas En(T), n ∈ ω.

Yeshkeyev А.Р. aibat.kz@gmail.com

The question of A.D.Taymanov

Thus the above mentioned question of A.D.Taymanov can be formulated as follows: (**) What properties must have lattice En, n ∈ ω that existed Jonsson theory T , such that En was isomorphic En(T), n ∈ ω? One can say that the problem (**) has a positive solution for Jonsson theory T, if there exists a sequence of lattices En, n ∈ ω, that En is isomorphic En(T), n ∈ ω.

Yeshkeyev А.Р. aibat.kz@gmail.com

The question of A.D.Taymanov

Theorem 3. Let T - perfect, complete of existential sentences ∆ − M-theory in the language of above signature and the theory T ∗

Π+ α+2

is ∆ − M-theory. Then the following conditions are equivalent: (1) a positive solution to the problem (*) with respect to the theory T c; (2) a positive solution (**) with respect to theory T ∗

Π+ α+2

, where T ∗ is the center of the theory T.

Yeshkeyev А.Р. aibat.kz@gmail.com

Perfectness

We consider the following result obtained in the language of the central type of this theory regarding to the perfectness in the frame

• f new signature.

Theorem 4. Let the theory T is complete for existential sentences ∆-J-theory in the language of the signature σΓ(A) = σ {ca| a ∈ A} Γ. Then the following conditions are equivalent:

1 T ∗ is perfect; 2 En(T) is Stone algebra. 3 En(T c) is Boolean algebra. Yeshkeyev А.Р. aibat.kz@gmail.com

Forcing companion

Standard defined forcing companion T f for ∆-M-theory T. It is forcing companion of the theory T. T f ={ϕ : T ϕ} Theorem 5. Let λ be an arbitrary infinite cardinal, T perfect ∆ − M-theory, complete for existential sentences. Then the following conditions are equivalent:

1 T ∗ is J − λ-stable; 2 (T ∗)F is λ - stable in the classical sense, where (T ∗)F is

forcing companion of theory T ∗ in enriched signature;

3 T c is λ - stable in the classical sense. Yeshkeyev А.Р. aibat.kz@gmail.com

Definition 5. Theory T is called convex if for any model A and for any family {Bi|i ∈ I} its substructures, which are models of the theory T, intersection

i∈I Bi is model of theory T. It is assumed that this

intersection is not empty. If this intersection is never empty, then the theory is said to be strongly convex. Definition 6. If the theory is strongly convex, then the intersection of all of its models is contained in some of its models.This model is called the core model of the theory. Definition 7. The model of the signature of this theory (hereinafter structure) is called core if it is isomorphic to a single substructure of each model

• f the theory.

Yeshkeyev А.Р. aibat.kz@gmail.com

Within the above definitions and in the enriched signature we have the following results. Let ∆ = B+(At). The assumption of a completeness of this theory is necessary due to the following fact. Lemma 1. In case of a positive Robinson’s theory, positive existential completeness implies ∆ − JEP, the converse is false.

Yeshkeyev А.Р. aibat.kz@gmail.com

Suppose that for considered theory whenever there is ϕ(x) an existential formula and prove in T, then there is some existential formula ψ(x) and an integer n, such that in T true ∃=nxϕ ∧ ∃x(ϕ ∧ ψ), and if T | = (δ1 ∨ δ2), where δ1, δ2 - some existential sentences, then T | = δ1 or T | = δ2. Theorem 6. Suppose that the theory T is ∆-R-perfect Jonsson strongly convex theory and it is positively existentially complete. Then the following conditions are equivalent: (1) theory T ∗ has a core structure; (2) theory T C is the core model.

Yeshkeyev А.Р. aibat.kz@gmail.com

Theorem 7. Let the theory T - ∆-R-strongly convex theory and perfect Jonsson, and it positively existentially complete. Then M a core structure of the theory T ∗ if and only if the model M is a core model of the center T ∗ in the above mentioned enrichment.

Yeshkeyev А.Р. aibat.kz@gmail.com

The following results are relevant to the description of "small"models in the class of existentially closed models convex positive Jonsson theories. By "small"we mean a countable prime countable and special atomic models for incomplete theories. We have a result concerning the syntactic and semantic conditions

• f the atomicity of the concepts of ∆-nice in the class ET theory T

with the following condition. Theorem 8. Suppose that the theory T - ∆-R-perfect almost closed Jonsson strongly convex theory and it is complete for positive ∀∃-sentences. A a countable model from ET. Then the following conditions are equivalent: (1) A − (∆, ∆)- atomic; (2) A ∈ ET and ∆-nice.

Yeshkeyev А.Р. aibat.kz@gmail.com

We say that a theory T admits R+

1 , if for any positive existential

formula ϕ (x) together with T there is a formula ψ (x) ∈ ∆+ consistent with T such that T | = ψ ↔ ϕ. Definition 8. Countable model of T is called countable algebraically universal model if in it can ∆-immersed all countable models of this theory. Model A is ∆-algebraically prime model of the theory T if A it is a model of the theory T and A can be ∆-immersed in every-model theory T.

Yeshkeyev А.Р. aibat.kz@gmail.com

Theorem 9 Suppose T– ∆-R-theory for the complete positive existential sentences, having a countable universal model. Then T there is an ∆-algebraically prime model, which is the (Σ, ∆+)-atomic. Theorem 10 Suppose T - ∆-R-complete theory for positive existential sentences, admitting R1. Then the following conditions are equivalent: (1) T has an ∆-algebraically prime model, (2) T has (Σ, ∆+)-atomic model, (3) T has a single ∆-algebraically prime model.

Yeshkeyev А.Р. aibat.kz@gmail.com

Let A, B ∈ (ET)+ and A ⊂

= B. Then B called ∆-algebraically prime

model extension A in (ET)+ if for any model C ∈ (ET)+ of what A ∆-immersed in C that B ∆-immersed in C. Theorem 11 Let T- ∆-R-theory complete for positive existential sentences for which the R+

1 is yields and ∆ = B(At). Then the following

conditions are equivalent: (1) T ∗ is ω1-categorical, (2) any countable model of (ET)+ has ∆-algebraically prime extension of the model in (ET)+.

Yeshkeyev А.Р. aibat.kz@gmail.com

Thank you very much for your attention!

Yeshkeyev А.Р. aibat.kz@gmail.com