The model theory of unitriangular groups

Annals of Pure and Applied Logic 68 (3):225-261 (1994)

Abstract
he model theory of groups of unitriangular matrices over rings is studied. An important tool in these studies is a new notion of a quasiunitriangular group. The models of the theory of all unitriangular groups are algebraically characterized; it turns out that all they are quasiunitriangular groups. It is proved that if R and S are domains or commutative associative rings then two quasiunitriangular 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 UTn are studied. If R is a skew field, they are of the form UTn, for some S ≡ 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 UTn ≡ UTn, but UTn cannot be represented in the form UTn for S ≡ R. We also study the number of models in a power of the theory of a unitriangular group. In particular, we prove that, for any communicative associative ring R and any infinite power λ, I = I). We construct an associative ring such that I = 3 and I) = 2. We also study models of the theory of UTn in the case of categorical R
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DOI 10.1016/0168-0072(94)90022-1
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References found in this work BETA

Model Theory of Modules.Martin Ziegler - 1984 - Annals of Pure and Applied Logic 26 (2):149-213.
Small Stable Groups and Generics.Frank O. Wagner - 1991 - Journal of Symbolic Logic 56 (3):1026-1037.
Small Stable Groups and Generics.Frank O. Wagner - 1991 - Journal of Symbolic Logic 56 (3):1026-1037.

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