Groups elementarily equivalent to a free nilpotent group of finite rank

Abstract

a b s t r a c tIn this paper, we give a complete algebraic description of groups elementarily equivalent to the P. Hall completion of a given free nilpotent group of finite rank over an arbitrary binomial domain. In particular, we characterize all groups elementarily equivalent to a free nilpotent group of finite rank.

Elementary coordinatization of finitely generated nilpotent groups

“…This paper continues the authors' efforts [11,12], in providing a comprehensive and uniform approach to various model-theoretical questions on algebras and nilpotent groups. We introduce our main techniques for approaching the following fundamental problems (stated here for nilpotent groups).…”

“…Consider the series (12) obtained in Lemma 8.4. Then there exists a pseudo-basis ū of G of length m adapted to (12) and there are natural numbers…”

Elementary equivalence of rings with finitely generated additive groups

“…This paper continues the authors' efforts [7,8,9], in providing a comprehensive and uniform approach to various model-theoretical questions on algebras and nilpotent groups. By a scalar ring we mean a commutative associative unitary ring.…”

Diophantine problems in solvable groups

“…This ring constitutes an important feature of f , and in some sense it provides an "approximation" to interpreting (in (A, B; f )) multiplication of constant elements from N and M by integer variables. It has been used successfully to study different first order theoretic aspects of different types of structures, including rings whose additive group is finitely generated [32], free algebras [22,23,24], and nilpotent groups [33,34]. For us the most important property of R(f ) is that it is e-interpretable in (A, B; f ): Theorem 3.1 (Theorem 3.5 of [16]).…”