kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2025-22-3-271-279kaz29-2122Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESОКРЕСТНОСТИ В СЛАБО УПОРОДЯЧЕННО МИНИМАЛЬНЫХ ТЕОРИЯХNEIGHBORHOODS IN WEAKLY ORDERED MINIMAL THEORIEShttps://orcid.org/0000-0002-3743-7404БайжановБ.BaizhanovB.

д.ф.-м.н., профессор

г. Алматы

г. Каскелен

Dr.Phys.-Math.Sc.

Almaty

Kaskelen 

baizhanov@math.kzhttps://orcid.org/0009-0005-3608-9259ТазабековаН.TazabekovaN.

докторант

г. Алматы

г. Каскелен

PhD candidate 

Almaty

Kaskelen 

tazabekova.nargiz@gmail.comhttps://orcid.org/0000-0001-7203-1701ЗамбарнаяТ.ZambarnayaT.

PhD

г. Алматы

PhD

Almaty

zambarnaya@math.kzИнститут математики и математического моделирования; Казахстанско-Британский технический университет; Университет СДУКазахстанInstitute of mathematics and mathematical modeling; Kazakh British Technical University; SDU UniversityKazakhstanИнститут математики и математического моделированияКазахстанInstitute of mathematics and mathematical modelingKazakhstan202527092025223271279Copyright © Байжанов Б., Тазабекова Н., Замбарная Т., 20252025Байжанов Б., Тазабекова Н., Замбарная Т.Baizhanov B., Tazabekova N., Zambarnaya T.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/2122

В данной работе исследуются окрестности, слабая ортогональность и почти ортогональность полных неалгебраических 1-типов в слабо упорядоченных минимальных (слабо о-минимальных) теориях. Окрестности вводятся как инструмент для описания локального поведения реализаций типов и обобщения понятия алгебраического замыкания внутри типа. Их использование позволяет выявить различия между типами и уточнить структуру их взаимодействия. Мы формулируем и доказываем основные свойства окрестностей. В частности, установлено, что  . На основе этих результатов изучаются соотношения между слабой ортогональностью и почти ортогональностью типов. В частности, получены критерии, описывающие их эквивалентность, симметричность и поведение для различных классов типов (иррациональных, квазисолитарных и квазирациональных). Таким образом, работа вносит вклад в уточнение и развитие понятий ортогональности для слабо о-минимальных теорий. Также показано, что для некоторых классов слабо о-минимальных теорий слабая и почти ортогональность совпадают. Полученные результаты предоставляют новые инструменты для анализа геометрии типов в слабо o-минимальных теориях и открывают перспективы применения в дальнейшем исследовании структур слабо о-минимального типа. Кроме того, предложенные подходы могут быть использованы для сопоставления с более общими классами теорий.

This paper studies neighborhoods, weak orthogonality, and almost orthogonality of complete non-algebraic 1-types in weakly ordered minimal (weakly o-minimal) theories. A neighborhood is introduced as a tool to describe the local properties of type realizations and to generalize the notion of algebraic closure within a type. Their use allows us to distinguish between types and to refine the structure of their interaction. We formulate and prove the main properties of neighborhoods. In particular, it was established that . On the basis of these results, we investigate the relationships between weak and almost orthogonality of types. In particular, we obtain criteria describing their equivalence, symmetry, and behavior for various classes of types (irrational, quasisolitary, and quasirational). Thus, the paper contributes to clarifying and developing the concepts of orthogonality in weakly o-minimal theories. It is also shown that for certain classes of weakly o-minimal theories, weak and almost orthogonality coincide. The results obtained provide new tools for analyzing the geometry of types in weakly o-minimal theories and open perspectives for further research on structures of weakly o-minimal type. In addition, the proposed approaches can be used for comparison with more general classes of theories.

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The authors declare that there are no conflicts of interest present.