The model theory of unitriangular groups
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 models in a power of the theory of a unitriangular group. In particular, he proves that, for any commutative associative ring \(R\) and any infinite power \(\lambda\), \(l(\lambda,R) = l(\lambda,{\mathbf U}{\mathbf T}_ n(R))\). He constructs an associative ring such that \(l(\aleph_ 1,R) = 3\) and \(l(\aleph_ 1,{\mathbf U}{\mathbf T}_ n(R)) = 2\). This solves a problem posed by \textit{C. R. Videla} [J. Pure Appl. Algebra 55, 289-302 (1988; Zbl 0658.03020)]. The author also studies models of the theory of \({\mathbf U}{\mathbf T}_ n(R)\) in the case of categorical \(R\).
model theory of groups of unitriangular \(n \times n\) matrices over rings
scientific article; zbMATH DE number 626271
scientific article