Complete first-order theories of some classical matrix groups over algebraic integers

Alexei G. Myasnikov; Mahmood Sohrabi

doi:10.1016/j.jalgebra.2021.04.015

Complete first-order theories of some classical matrix groups over algebraic integers

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Abstract

Let O be the ring of integers of a number field, and let n≥3. This paper studies bi-interpretability of the ring of integers Z with the special linear group SL_n(O), the general linear group GL_n(O) and the subgroup T_n(O) of GL_n(O) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

Original language	English
Pages (from-to)	206-231
Number of pages	26
Journal	Journal of Algebra
Volume	582
DOIs	https://doi.org/10.1016/j.jalgebra.2021.04.015
State	Published - 15 Sep 2021

Keywords

Abelian deformation
Bi-interpretability
Elementary equivalence
General linear group
Special linear group

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10.1016/j.jalgebra.2021.04.015

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title = "Complete first-order theories of some classical matrix groups over algebraic integers",

abstract = "Let O be the ring of integers of a number field, and let n≥3. This paper studies bi-interpretability of the ring of integers Z with the special linear group SLn(O), the general linear group GLn(O) and the subgroup Tn(O) of GLn(O) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.",

keywords = "Abelian deformation, Bi-interpretability, Elementary equivalence, General linear group, Special linear group",

author = "Myasnikov, \{Alexei G.\} and Mahmood Sohrabi",

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year = "2021",

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T1 - Complete first-order theories of some classical matrix groups over algebraic integers

AU - Myasnikov, Alexei G.

AU - Sohrabi, Mahmood

PY - 2021/9/15

Y1 - 2021/9/15

N2 - Let O be the ring of integers of a number field, and let n≥3. This paper studies bi-interpretability of the ring of integers Z with the special linear group SLn(O), the general linear group GLn(O) and the subgroup Tn(O) of GLn(O) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

AB - Let O be the ring of integers of a number field, and let n≥3. This paper studies bi-interpretability of the ring of integers Z with the special linear group SLn(O), the general linear group GLn(O) and the subgroup Tn(O) of GLn(O) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

KW - Abelian deformation

KW - Bi-interpretability

KW - Elementary equivalence

KW - General linear group

KW - Special linear group

UR - https://www.scopus.com/pages/publications/85105603995

UR - https://www.scopus.com/pages/publications/85105603995#tab=citedBy

U2 - 10.1016/j.jalgebra.2021.04.015

DO - 10.1016/j.jalgebra.2021.04.015

M3 - Article

AN - SCOPUS:85105603995

SN - 0021-8693

VL - 582

SP - 206

EP - 231

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Complete first-order theories of some classical matrix groups over algebraic integers

Abstract

Keywords

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