# Model-theory of vector-spaces

over unspecified fields

This paper appeared in the *Archive for Mathematical Logic,* vol. 48, no
5 (2009), pp. 421–436, DOI 10.1007/s00153-009-0130-x

**Abstract.**
Vector-spaces over unspecified fields can be axiomatized as one-sorted
structures, namely, abelian groups with the relation of parallelism.
Parallelism is binary linear dependence. When equipped with the
*n*-ary relation of linear dependence for some positive integer
*n*, a vector-space is existentially closed if and only if it is
*n*-dimensional over an algebraically closed field. In the
signature with an *n*-ary predicate for linear dependence for
*each* positive integer *n*, the theory of
infinite-dimensional vector-spaces over algebraically closed fields is
the model-completion of the theory of vector-spaces.