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.