On definability of types and relative stability

Verbovskiy, Viktor

doi:10.1002/malq.201600084

Cited by 4 publications

(1 citation statement)

References 15 publications

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“…V.V. Verbovskiy continued to study relatively stable theories and definability of types in [344] and proved that for a theory T that is stable up to ∆ it holds that any one-type over a model of T is definable if and only if its ∆-part is definable. In [342] (2018) V. Verbovskiy continued to study o-stable ordered groups and proved that any ordered group of Morley o-rank 1 with boundedly many definable convex subgroups is weakly o-minimal and constructed an example of an ordered group of Morley o-rank 1 and Morley o-degree at most 4.…”

Section: B2 On the One Hand If For Somementioning

confidence: 99%

A. D. Taimanov and model theory in Kazakhstan

Baizhanov¹,

Kulpeshov²,

Zambarnaya³

2020

Sib. Elektron. Mat. Izv.

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Section: B2 On the One Hand If For Somementioning

confidence: 99%

A. D. Taimanov and model theory in Kazakhstan

Baizhanov¹,

Kulpeshov²,

Zambarnaya³

2020

Sib. Elektron. Mat. Izv.

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On o-Stable Expansions of $$\boldsymbol{(\mathbb{Z},<,+)}$$

Verbovskiy,

Yershigeshova

2023

Lobachevskii J Math

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Examples of Linear Orders With a Definable Unary Function and the Independence Property

Verbovskiy,

Yershigeshova

2023

jour

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After the appearance of the concept of o-minimality, which was introduced by L. van den Dries for expansions of the ordered field of real numbers and generalized to arbitrary linear orders by A. Pillay and C. Steinhorn, linearly ordered structures became firmly established in the circle of interests of specialists in model theory. Numerous generalizations of the concept of o-minimality have appeared in the works of various authors, such as weak o-minimality, quasi-o- minimality, weak quasi-o-minimality, dp-minimality, and o-stability. B. S. Baizhanov and V. V. Verbovskiy proved that o-stability generalizes all the above concepts for linearly ordered structures and that o-stability entails the absence of the independence property. They also proved that any linear order has an o-superstable theory. V. V. Verbovskiy studied o-stable ordered groups, in particular, he proved that they are commutative. In this paper, we begin the study of the question of how complex the theory of a linear order with one unary function can be. We construct an example of an expansion of a linearly ordered structure with one unary function, which has the independence property.

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On definability of types and relative stability

Cited by 4 publications

References 15 publications

A. D. Taimanov and model theory in Kazakhstan

A. D. Taimanov and model theory in Kazakhstan

On o-Stable Expansions of $$\boldsymbol{(\mathbb{Z},<,+)}$$

Examples of Linear Orders With a Definable Unary Function and the Independence Property

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