Divisible Rigid Groups. Morley Rank

N. S. Romanovskii

doi:10.1007/s10469-022-09689-5

Divisible Rigid Groups. Morley Rank

Переведенное название: Делимые жёсткие группы. Ранг Морли

N. S. Romanovskii

Кафедра алгебры и математической логики ММФ

Результат исследования: Научные публикации в периодических изданиях › статья › рецензирование

Аннотация

Пусть G - счётная насыщенная модель теории Tm делимых m-жёстких групп и зафиксировано расщепление G1G2…Gm группы G в полупрямое произведение абелевых подгрупп. Сопоставим каждому набору (n1,…,nm) неотрицательных целых чисел ординал α=ωm−1nm+…+ωn2+n1 и обозначим через G(α) множество Gn11×Gn22×…×Gnmm, оно определимо над G в Gn1+…+nm. Тогда ранг Морли множества G(α) относительно G равен α. Отсюда следует, что RM(G)=ωm−1+ωm−2+…+1.

Переведенное название	Делимые жёсткие группы. Ранг Морли
Язык оригинала	английский
Страницы (с-по)	207-224
Число страниц	18
Журнал	Algebra and Logic
Том	61
Номер выпуска	3
DOI	https://doi.org/10.1007/s10469-022-09689-5
Состояние	Опубликовано - июл. 2022

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title = "Divisible Rigid Groups. Morley Rank",

abstract = "Let G be a countable saturated model of the theory Im of divisible m-rigid groups. Fix the splitting G1G2..Gm of a group G into a semidirect product of Abelian groups. With each tuple (n1,.. , nm) of nonnegative integers we associate an ordinal α = ωm−1nm+.. + ωn2 + n1 and denote by G(α) the set G1n1×G2n2×…×Gmnm, which is definable over G in Gn1+…+nm. Then the Morley rank of G(α) with respect to G is equal to α. This implies that RM (G) = ωm−1 + ωm−2 +.. + 1.",

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author = "Romanovskii, {N. S.}",

note = "Funding Information: This work was supported by the National Natural Science Foundation of China (No. 81971406, 81871185), The 111 Project (Yuwaizhuan (2016)32), Chongqing Science & Technology Commission (cstc2021jcyj-msxmX0213), Chongqing Municipal Education Commission (KJZD-K202100407), Chongqing Health Commission and Chongqing Science & Technology Commission (2021MSXM121, 2020MSXM101). Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.",

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N2 - Let G be a countable saturated model of the theory Im of divisible m-rigid groups. Fix the splitting G1G2..Gm of a group G into a semidirect product of Abelian groups. With each tuple (n1,.. , nm) of nonnegative integers we associate an ordinal α = ωm−1nm+.. + ωn2 + n1 and denote by G(α) the set G1n1×G2n2×…×Gmnm, which is definable over G in Gn1+…+nm. Then the Morley rank of G(α) with respect to G is equal to α. This implies that RM (G) = ωm−1 + ωm−2 +.. + 1.

AB - Let G be a countable saturated model of the theory Im of divisible m-rigid groups. Fix the splitting G1G2..Gm of a group G into a semidirect product of Abelian groups. With each tuple (n1,.. , nm) of nonnegative integers we associate an ordinal α = ωm−1nm+.. + ωn2 + n1 and denote by G(α) the set G1n1×G2n2×…×Gmnm, which is definable over G in Gn1+…+nm. Then the Morley rank of G(α) with respect to G is equal to α. This implies that RM (G) = ωm−1 + ωm−2 +.. + 1.

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