Viktor Verbovskiy | Süleyman Demirel University (original) (raw)

Papers by Viktor Verbovskiy

Given a cut s in an ordered structure M we can define a localization of Morley rank-Morley o-rank... more Given a cut s in an ordered structure M we can define a localization of Morley rank-Morley o-rank, replacing each formula in definition of Morley rank with the following partial types: the cut s extended with this formula. We prove in the paper that any ordered group of Morley o-rank 1 with boundedly many definable convex subgroups is weakly o-minimal and construct an example of an ordered group of Morley o-rank 1 and Morley o-degree at most 4.

Abstract. A theory is stable up to ∆ if any ∆-type over a model has a few extensions up to comple... more Abstract. A theory is stable up to ∆ if any ∆-type over a model has a few extensions up to complete types. I prove that a theory has no the independence property iff it is stable up to some ∆, where each ϕ(x; ¯y) ∈ ∆ has no the independence property. Definability of one-types over a model of a stable up to ∆ theory is investigated. 1.

Abstract. An ordered structure M is called o-λ-stable if for any subset A with |A | ≤ λ and for a... more Abstract. An ordered structure M is called o-λ-stable if for any subset A with |A | ≤ λ and for any cut in M there are at most λ 1-types over A which are consistent with the cut. It is proved in the paper that an ordered o-stable group is abelian. Also there were investigated definable subsets and unary functions of o-stable groups.

An ordered structure M is called o-λ-stable if for any subset A with |A| ≤ λ and for any cut in M... more An ordered structure M is called o-λ-stable if for any subset A with |A| ≤ λ and for any cut in M there are at most λ 1-types over A which are consistent with the cut. It is proved in the paper that an ordered o-stable group is abelian. Also there were investigated definable subsets and unary functions of o-stable groups.

The purpose of this research is to examine the effects of problem posing intervention on 8th grad... more The purpose of this research is to examine the effects of problem posing intervention on 8th grade students’ mathematics achievement and attitudes toward mathematics. Word problems were used in the research as a tool to observe the differences between experimental and control groups. We analyzed the effects of problem posing instruction by specially designed tests on pre and post activities. Meanwhile we sought student responses through individual meetings.This study has been conducted with 8th grade students at a Kazakh High School for gifted students during the second semester of 2010-2011 academic years. There were 54 students in total that were divided into two groups. One of the groups was experimental and the other was control group. There was equal number of students in each group with a number of 27. The research took two months in the same school.The research used a mixed methods design with quantitative and qualitative components. Data from quantitative component that was ...

Let M be strongly minimal and constructed by a ‘Hrushovski construction’. If the Hrushovski algeb... more Let M be strongly minimal and constructed by a ‘Hrushovski construction’. If the Hrushovski algebraization function μ is in a certain class T (μ triples) we show that for independent I with |I| > 1, dcl(I) = ∅ (* means not in dcl of a proper subset). This implies the only definable truly n-ary function f (f ‘depends’ on each argument), occur when n = 1. We prove, indicating the dependence on μ, for Hrushovski’s original construction and including analogous results for the strongly minimal k-Steiner systems of Baldwin and Paolini that the symmetric definable closure, sdcl(I) = ∅, and thus the theory does not admit elimination of imaginaries. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if k = p. The proofs depend on our introduction for appropriate G ⊆ aut(M) the notion of a Gnormal substructure A of M and of a G-decomposition of A. These results lead to a finer classificat...

Mathematical Logic Quarterly

International Online Journal of Primary Education Issn 1300 915x, May 26, 2014

An ordered structure M is called o- -stable if for any subset A with |A| and for any cut in M the... more An ordered structure M is called o- -stable if for any subset A with |A| and for any cut in M there are at most 1-types over A which are consistent with the cut. It is proved in the paper that an ordered o-stable group is abelian. Also there were investigated definable subsets and unary functions of o-stable groups.

It has long been known [K. Urbanik, Fundam. Math. 57, 215–236 (1965; Zbl 0171.28103)] that any gr... more It has long been known [K. Urbanik, Fundam. Math. 57, 215–236 (1965; Zbl 0171.28103)] that any group could be represented in a strongly minimal theory by just writing down the relations of the group as unary functions. We show the same process works for ordered groups and yields an o-minimal group.

Annals of Pure and Applied Logic, 2003

A totally ordered group G (possibly with extra structure) is called coset-minimal if every defina... more A totally ordered group G (possibly with extra structure) is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G {±∞}. Continuing work in Belegradek et al. (J. Symbolic Logic 65(3) (2000) 1115) ...

Algebra and Logic, 2011

A well-developed technique created to study stable theories (M. Morley, S. Shelah) is applied in ... more A well-developed technique created to study stable theories (M. Morley, S. Shelah) is applied in dealing with a class of theories with definable linear order. We introduce the notion of an o-stable theory, which generalizes the concepts of o-minimality, of weak o-minimality, and of quasi-o-minimality. It is proved that o-stable theories are dependent, but they do not exhaust the class of dependent theories with definable linear order, and that every linear order is o-superstable.

Annals of Pure and Applied Logic, 2003

Mathematical Logic Quarterly, 2015

ABSTRACT Here we study properties of weakly circularly minimal cyclically ordered groups. The mai... more ABSTRACT Here we study properties of weakly circularly minimal cyclically ordered groups. The main result of the paper is that any weakly circularly minimal cyclically ordered group is abelian.

In the paper [MMS] some basic properties of weakly o-minimal structures have been investigated. I... more In the paper [MMS] some basic properties of weakly o-minimal structures have been investigated. In this work we introduce new notions finite depth and strong monotonicity with the help of which we define new necessary conditions of weak o-minimality of elementary theory of a weakly o-minimal structure. We adduce in the second part examples of weakly o-minimal structures without weakly o-minimal theories with infinite and finite depths. Note, that the existence of these structures has been announced in [MMS].

Èíñòèòóò ïðîáëåì èíôîðìàòèêè è óïðàâëåíèÿ ÌÎÍ ÐÊ 480100 Àëìàòû, Ïóøêèíà óë., 125, vvv@ipic.kz Ïóñ... more Èíñòèòóò ïðîáëåì èíôîðìàòèêè è óïðàâëåíèÿ ÌÎÍ ÐÊ 480100 Àëìàòû, Ïóøêèíà óë., 125, vvv@ipic.kz Ïóñòü H ïëîòíàÿ ïîäãðóïïà â R. Ìû äîêàaeåì ÷òî òîãäà (R, <, +, 0, H, H q , e p,q,i ) q∈Q + äîïóñêàåò ýëèìèíàöèþ êâàíòîðîâ, ãäå H q = {q · h : h ∈ H} äëÿ íåêîòîðîãî q ∈ Q + . Åñëè äëÿ íåêîòîðûõ p, q ∈ Q + èìååò ìåñòî |H q :

A simplified proof of B. S. Baizhanov’s following theorem [J. Symb. Log. 66, No. 3, 1382–1414 (20... more A simplified proof of B. S. Baizhanov’s following theorem [J. Symb. Log. 66, No. 3, 1382–1414 (2001; Zbl 0992.03047)] is given: An expansion of a model of a weakly o-minimal theory by a unary convex predicate has a weakly o-minimal theory. Here, a totally ordered structure is called weakly o-minimal if every parametrically definable subset is the finite union of convex sets, and a theory is called weakly o-minimal if every one of its models is weakly o-minimal.

Algebra and model theory

In the paper [2] the following problem was put: "If M is a weakly ominimal expansion of an ordere... more In the paper [2] the following problem was put: "If M is a weakly ominimal expansion of an ordered group, must it have weakly o-minimal theory?" Here we construct a counterexample.

Annals of Pure and Applied Logic, Jan 1, 2003

Continuing work of Baldwin and Shi (Ann. Pure Appl. Logic 79 (1996) 1), we study non-!-saturated ... more Continuing work of Baldwin and Shi (Ann. Pure Appl. Logic 79 (1996) 1), we study non-!-saturated generic structures of the ab initio Hrushovski construction with amalgamation over closed sets. We show that they are CM-trivial with weak elimination of imaginaries. Our main tool is a new characterization of non-forking in these theories. This was generalized by Baudisch , who constructed a new ℵ 1 -categorical CMtrivial group with weak elimination of imaginaries by replacing the number of tuples satisfying a relation in the ab initio construction by the dimension of an appropriate vector space. The ab initio construction yields important examples as in . All the generic structures mentioned above are !-saturated. On the other hand, non-!saturated generic structures in a ÿnite relational language have been studied by Baldwin, Shelah and Shi, who show that they are stable [4] and near model-complete . There are many papers related to generic structures, for example ; for further results see .

Annals of Pure and Applied …, Jan 1, 2003

A totally ordered group G (possibly with extra structure) is called coset-minimal if every deÿnab... more A totally ordered group G (possibly with extra structure) is called coset-minimal if every deÿnable subset of G is a ÿnite union of cosets of deÿnable subgroups intersected with intervals with endpoints in G ∪ {±∞}. Continuing work in Belegradek et al. (J. Symbolic Logic 65(3) (2000) 1115) and Point and Wagner (Ann. Pure Appl. Logic 105(1-3) ( , we study coset-minimality, as well as two weak versions of the notion: eventual and ultimate coset-minimality. These groups are abelian; an eventually coset-minimal group, as a pure ordered group, is an ordered abelian group of ÿnite regular rank. Any pure ordered abelian group of ÿnite regular rank is ultimately coset-minimal and has the exchange property; moreover, every deÿnable function in such a group is piecewise linear. Pure coset-minimal and eventually coset-minimal groups are classiÿed. In a discrete coset-minimal group every deÿnable unary function is piece-wise linear (this improves a result in Point and Wagner (Ann. Pure Appl. Logic 105(1-3) , where coset-minimality of the theory of the group was required). A dense coset-minimal group has the exchange property (which is false in the discrete case (M.S.R.I., preprint series, 1998-051)); moreover, any deÿnable unary function is piecewise

Mathematical Logic Quarterly

International Online Journal of Primary Education Issn 1300 915x, May 26, 2014

Annals of Pure and Applied Logic, 2003

Algebra and Logic, 2011

Annals of Pure and Applied Logic, 2003

Mathematical Logic Quarterly, 2015

Algebra and model theory

Annals of Pure and Applied Logic, Jan 1, 2003

Annals of Pure and Applied …, Jan 1, 2003