Here is some naive questions that came to one’s mind when considering vector spaces where the dimension is not finite but orbit-finite: under a reasonable assumption on atoms , say -categoricity,

— is it true that every vector subspace of an orbit-finite dimensional vector space is orbit-finite dimensional?

— is it true that the set of solutions of an orbit-finite  system of linear equations is orbit-finite dimensional?

Let be a field (think of rationals) and let be an orbit finite set over some atoms . Denote by

the set of all functions from to that are almost everywhere 0: for almost all , . We can call such functions vectors of dimension . Clearly has the structure of vector space.

A vector space we call orbit-finite dimensional if for some orbit-finite subset . In words, is spanned by its orbit-finite subset . Clearly is orbit-finite dimensional when is orbit-finite, as it  is spanned by the set of unit vectors.

Question 1: under a reasonable assumption on atoms , say -categoricity, is it true that every vector subspace of an orbit-finite dimensional vector space is orbit-finite dimensional?

Systems of linear equations can be used to define vector subspaces. Consider and an -indexed family of vectors of dimension :

where and are arbitratry orbit-finite sets. Think of as the set of columns of a matrix , where is the column of indexed by ; and think of as the set of rows of . Thus we can think of as of an orbit-finite system of linear equations, which defines the following set of solutions:

( stands for the zero vector in ). Note that the sum is well defined as for only finitely many . Clearly is a vector subspace of .  If Quesion 1 has positive answer, it is orbit-finite dimensional. Otherwise, it makes sense to ask:

Question 2: under a reasonable assumption on atoms , say -categoricity, is it true that the set of solutions of an orbit-finite  system of linear equations is orbit-finite dimensional?