Университеттің 85 жылдығына арналған Қазіргі заманғы математика
EXPANSION OF A MODEL BY UNARY EXTERNALLY DEFINABLE SET AND
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| EXPANSION OF A MODEL BY UNARY EXTERNALLY DEFINABLE SET AND
NEIGHBORHOOD OF ELEMENT IN TYPE B.S.BAIZHANOV AND F.SARGULOVA Institute of Mathematics and Mathematical Modeling External definability. Let ɱ be elementary substructure of ɲ. It is said that pair of models is beautiful, if ɲ is saturated over M. Let α ¯ ∈ N \ M and p := tp ( α | M ). Then for any formula ψ ( x ¯ ,y ¯) define the predicate R ( ψ,p )( y ¯) on the set M ,|= R ( ψ,p )( a ¯) i ff ψ ( x ¯ ,a ¯) ∈ tp ( α ¯| M ) i ff ɲ|= ψ ( α ¯ ,a ¯). Denote by ɱ + = ⟨ M ;Σ + ⟩ where Σ + := { R ( ψ,p )( y ¯) | p ∈ S ( M ) ,ψ ∈ Σ}. Let ɱ be a model of an arbitrary complete theory T of the signature Σ. We say that ɱ + is expansion of ɱ by type p ∈ S 1 ( M ), if ɱ + := ⟨ M ;Σ + ⟩ , where ∑ + :={ R ( ψ.p )( y ¯)| ψ ∈ Σ}. p p We say that ɱ + admits uniformly representation of ∑ + -formulas by Σ-formulas, if for p p any formula ϕ ( y ¯) of ∑ + there exists Σ-formula K ϕ ( y ¯ ,z ¯), there exists α ¯ ∈ N \ M such that for any a ¯ ∈ M the following holds: ɱ + |= ϕ ( a ¯) ⇐⇒ |= K ϕ ( a ¯ ,α ¯) . Approach of Macpherson-Marker-Steinhorn. In the paper [1] (preprint 1994 Macpherson- Marker-Steinhorn proved weak o-minimality of the expansion of an o-minimal structure by unary convex predicate, such that the predicate is traversed by a uniquely realizable 1-type. Following D. Marker [5], an uniquely realizable 1-type p ∈ S 1 ( M ) over model is that prime model over model and one realization of this 1-type p contains just this element from the set of realization of the type. An uniquely realizable 1-type has the next property: there is no definable functionacting on the set of realizations of this 1-type p . Macpherson-Marker-Steinhorn considered at the same time two structures ɱ + = ⟨ M ;Σ ∪ {U 1 } ⟩ and ɲ = ⟨ M ;Σ ⟩ , where ɲ is a model of an o-minimal theory of the signature Σ and a saturated elementary extension of ɱ . They defined a new unary convex predicate U by using an element α ∈ N \ M from the set of realizations of an irrational 1−type p ∈ S 1 ( M ) such that for every a ∈ M the following holds: ɱ + |= U ( a ) ⇔ ɲ|= a<α . Thus, any Σ + - M -1-formula ϕ ( x,a ¯) has the set of its realizations, ϕ ( ɱ + ,a ¯)= K ϕ (ɲ ,a ¯) ∩ M , being a finite union of convex sets because K ϕ (ɲ ,a ¯) is a finite union of intervals and points. The elementary theory of ɱ + is weakly o- minimalsince the number of convex sets is bounded and consequently does not depend on parameters. Approach of B.S.Baizhanov. For the case when p ∈ S 1 ( M ) is a non uniquely realizable type, B.S. Baizhanov proposed [2] (1995), on the base of theory of (non)orthogonality of 1-types |