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

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Computation with atoms