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The model theory of unitriangular groups

Oleg V. Belegradek

1994
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Annals of Pure and Applied Logic
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The 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 (of fixed nilpotency class) 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
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... antiisomorphic. This algebraic result is new even for ordinary unitriangular groups. The groups elementarily equivalent to a single unitriangular group UT"(R) are studied. If R is a skew field, they are of the form UT,(S), 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 UT"(P) = UT,(R), but UT,(P) cannot be represented in the form UT,(S) 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 1, 1(1, R) = I(,?, UT"(R)). We construct an associative ring such that I@, , R) = 3 and I(K1 ,UT,(R)) = 2. We also study models of the theory of UT,(R) in the case of categorical R. For an associative ring with unit R, let UT,(R) be the group of all upper unitriangular matrices over R, that is matrices with entries in R which have zeros below the main diagonal and units on it. For n = 1 the group is trivial and for n = 2 it is isomorphic to the additive group of R; so the only interesting case is n 3 3. For any n, the group UT,(R) is (n -1) step nilpotent. The model theory of unitriangular groups began with Maltsev's paper [16]. He considered only the case n = 3. (Note that UT,(R) is a group even if R is not associative.) He showed that the ring R can be interpreted in the group UT,(R) with certain parameters and gave an algebraic characterization of groups of this form. Rose Cl93 applied the idea of Maltsev's work to the ring NT"(R) of all upper niltriangular n x n matrices, n > 3. He showed that the ring R is interpretable in the ring NT,(R) with parameters and gave a first order axiomatization of the class of all rings of the from NT"(F), where F is a field. It follows that every ring elementarily *

doi:10.1016/0168-0072(94)90022-1
fatcat:xynxr5q3pngnnmybuosai6ramq