Orbit-finite dimensional vector spaces, and orbit-finite systems of linear equations

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 \mathbb A, say \omega-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 \mathbb Q be a field (think of rationals) and let \cal X be an orbit finite set over some atoms \mathbb A. Denote by

    \[ {\mathbb Q}^{\cal X} \quad = \quad  {\cal X} \stackrel{0}{\to} {\mathbb Q} \]

the set of all functions v from \cal X to \mathbb Q that are almost everywhere 0: for almost all x\in {\cal X}, v(x) = 0. We can call such functions vectors of dimension \cal X. Clearly {\mathbb Q}^{\cal X} has the structure of vector space.

A vector space \cal V we call orbit-finite dimensional if {\cal V} = \text{Lin}(\cal Y) for some orbit-finite subset {\cal Y} \subseteq {\cal V}. In words, \cal V is spanned by its orbit-finite subset \cal Y. Clearly {\mathbb Q}^{\cal X} is orbit-finite dimensional when \cal X is orbit-finite, as it  is spanned by the set of unit vectors.

Question 1: under a reasonable assumption on atoms \mathbb A, say \omega-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 {\mathbb Q}^{\cal X} and an \cal X-indexed family of vectors of dimension \cal Y:

    \[ M \quad = \quad (M(x) \in {\mathbb Q}^{\cal Y})_{x\in\cal X}, \]

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

    \[ \text{sol}(M) \quad = \quad \{ s \in {\mathbb Q}^{\cal X} \ | \ \sum_{x\in\cal X} s(x) \cdot M(x) = \vec 0 \} \]

(\vec 0 stands for the zero vector in {\mathbb Q}^{\cal Y}). Note that the sum is well defined as s(x) \neq 0 for only finitely many x\in\cal X. Clearly \text{sol}(M) is a vector subspace of {\mathbb Q}^{\cal X}.  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 \mathbb A, say \omega-categoricity, is it true that the set of solutions of an orbit-finite  system of linear equations is orbit-finite dimensional?

 

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