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 of variables. L + = Q ( B + ( At ) is the set of formulas in prenex normal type obtained 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 h 1 : A → ∆ C , h 2 : B → ∆ C . Definition 2. The theory T admits ∆ − AP , if for any A , B , C ∈ ModT with h 1 : A → ∆ C , g 1 : A → ∆ B , where h 1 , g 1 are ∆ -homomorphism’s, there are exist D ∈ Modt and h 2 : C → ∆ D , g 2 : B → ∆ D where h 2 , g 2 are ∆ -homomorphism’s such that h 2 ◦ h 1 = g 2 ◦ g 1 . 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 T is 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 | = ∃ ¯ y ϕ ( h (¯ a ) , ¯ y ) ⇒ A | = ∃ ¯ y ϕ (¯ 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 σ Γ ( A ) = σ � { c a | a ∈ A } � Γ , where Γ = { g } � { c } � { P } . Consider the following theory T PgM ( A ) = Th Π + α +2 ( C , a ) a ∈ A � { g ( a ) = a | a ∈ Γ � { P ( c ) } � { ” P ⊆ ” } , where T g - expresses the A } � g ( c ) � T g fact that for any model ( M , g M ) | = T g holds: (1) g M -automorphism of M; (2) there is { m ∈ M | g M ( 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 σ Γ , 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) F n ( T ) , n ∈ ω , theory Т . In connection with these Boolean algebras F n ( T ) , n ∈ ω , is well-known question A.D. Taimanov : (*) What properties must have Boolean algebras B n , n ∈ ω that there exists a complete theory T , so that B n was isomorphic F n ( 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 ∼ = F 1 ( 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 B 1 , B 2 there is a categorical in a countable power theory T , such that F 1 ( T ) ∼ = B 1 , F 2 ( T ) ∼ = B 2 is necessary and sufficient that the number of atoms of B 2 was greater than the square of the number of atoms B 1 . 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 F n ( T ) , n ∈ ω , should be considered lattice existential formulas E n ( 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 E n , n ∈ ω that existed Jonsson theory T , such that E n was isomorphic E n ( T ) , n ∈ ω ? One can say that the problem (**) has a positive solution for Jonsson theory T , if there exists a sequence of lattices E n , n ∈ ω , that E n is isomorphic E n ( 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 ∗ is Π+ α +2 ∆ − 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 ∗ , where T ∗ Π+ α +2 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 of new signature. Theorem 4. Let the theory T is complete for existential sentences ∆ -J-theory in the language of the signature σ Γ ( A ) = σ � { c a | a ∈ A } � Γ . Then the following conditions are equivalent: 1 T ∗ is perfect; 2 E n ( T ) is Stone algebra. 3 E n ( 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 { B i | i ∈ I } its substructures, which are models of the theory T , intersection � i ∈ I B i 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 of the theory. Yeshkeyev А.Р. aibat.kz@gmail.com
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