Divisible Rigid Groups. Morley Rank

N. S. Romanovskii

doi:10.1007/s10469-022-09689-5

Divisible Rigid Groups. Morley Rank

N. S. Romanovskii

Section of Algebra and Mathematical Logic

Research output: Contribution to journal › Article › peer-review

Abstract

Let G be a countable saturated model of the theory I_m of divisible m-rigid groups. Fix the splitting G₁G₂..G_m of a group G into a semidirect product of Abelian groups. With each tuple (n₁,.. , n_m) of nonnegative integers we associate an ordinal α = ω^m−1n_m+.. + 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) = ω^m−1 + ω^m−2 +.. + 1.

Translated title of the contribution	Делимые жёсткие группы. Ранг Морли
Original language	English
Pages (from-to)	207-224
Number of pages	18
Journal	Algebra and Logic
Volume	61
Issue number	3
DOIs	https://doi.org/10.1007/s10469-022-09689-5
Publication status	Published - Jul 2022

Keywords

divisible m-rigid group
Morley rank

OECD FOS+WOS

1.01 MATHEMATICS

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10.1007/s10469-022-09689-5

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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.",

keywords = "divisible m-rigid group, Morley rank",

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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AU - Romanovskii, N. S.

N1 - 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: © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/7

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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.

KW - divisible m-rigid group

KW - Morley rank

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DO - 10.1007/s10469-022-09689-5

M3 - Article

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VL - 61

SP - 207

EP - 224

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 3

ER -

Divisible Rigid Groups. Morley Rank

Abstract

Keywords

OECD FOS+WOS

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