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?