The model theory of unitriangular groups
type: Item
instance of: scholarly article
title: The model theory of unitriangular groups
zbMATH Open document ID: 0807.03025
DOI: 10.1016/0168-0072(94)90022-1
author: Oleg Belegradek
published in: Annals of Pure and Applied Logic
publication date: 1995-03-05T00:00:00Z
review text: In the paper under review the model theory of groups of unitriangular \(n \times n\) matrices over rings is studied. The starting point of these studies was the paper of \textit{A. I. Mal'tsev} [Mat. Sb., Nov. Ser. 50, 257-266 (1960; Zbl 0100.014)] who showed how to interprete the ring \(R\) in the group \(\text{UT}_ 3(R)\) and gave an algebraic characterization of groups of this form. The studies of the case \(n = 3\) have been continued in a series of the author's papers, in particular in ``The Mal'tsev correspondence revisited'' [in: Algebra, Proc. Int. Conf. Memory A. I. Mal'tsev, Novosibirsk/USSR 1989, Contemp. Math. 131, Pt. 1, 37-59 (1992; Zbl 0754.03027)] and in ``Quasiunitriangular groups'' [J. Symb. Logic 58, 205-218 (1993; Zbl 0791.03016)]. In the paper under review the author studies the general case of arbitrary \(n \geq 3\). An important tool is a new notion of a quasiunitriangular group. The models of the theory of all \(\text{UT} _ n\)-groups are algebraically characterized; it turns out that all are quasi-\(\text{UT}_ n\)-groups. It is proved that if \(R\) and \(S\) are domains or commutative rings, then two quasi-\(\text{UT}_ n\)-groups groups over \(R\) and \(S\) are isomorphic only if \(R\) and \(S\) are isomorphic or antiisomorphic. This algebraic result is new even for ordinary unitriangular groups. The groups elementarily equivalent to a single unitriangular group \({\mathbf U}{\mathbf T}_ n(R)\) are studied. If \(R\) is a skew field, they are of the form \({\mathbf U}{\mathbf T}_ n(S)\), for some \(S \equiv R\). In general, the situation is not so nice. Examples are constructed demonstrating that such a group need not be a unitriangular group over some ring; moreover, there are rings \(P\) and \(R\) such that \({\mathbf U}{\mathbf T}_ n(P) \equiv {\mathbf U}{\mathbf T}_ n(R)\), but \({\mathbf U}{\mathbf T}_ n(P)\) cannot be represented in the form \({\mathbf U}{\mathbf T}_ n(S)\), for \(S \equiv R\). The author also studies the number of mode…
reviewed by: Oleg Belegradek
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20H25
zbMATH DE Number: 626271
zbMATH Keywords: model theory of groups of unitriangular \(n \times n\) matrices over rings
zbMATH Keywords: unitriangular groups
MaRDI profile type: Publication
full work available at URL: 0168-0072(94)90022-1
OpenAlex ID: W2019866004
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description: scientific article; zbMATH DE number 626271