Isomorphism and elementary equivalence of multilinear maps

Abstract

In [14], we proved that two finitely generated finite-by-nilpotent groups G,H are elementarily equivalent if and only if Z×G and Z×H are isomorphic. In the present paper, we obtain similar characterizations of elementary equivalence for the following classes of structures: 1. the (n+2)-tuples (A 1…,A n+1,f),where n≥2 is an integerA 1…,A n+1 are disjoint finitely generated abelian groups and f A 1×…×A n →A n+1: is a n-linear map; 2. the triples (A,B f), where n≥2 is an integerA,B are disjoint finitely generated abelian groups and f : A n →B is a n-linear map; 3. the couples (A,f), where n≥2 is an integerA is a finitely generated abelian group and f:A n →A is a n-linear map. For each class, we show that elementary equivalence does not imply isomorphism. In particular, we give an example of two nonisomorphic finitely generated torsion-free Lie rings which are elementarily equivalent.