Orbits of maximal vector spaces

Let $V_\infty$ be a standard computable infinite-dimensional vector space over the field of rationals. The lattice $\mathcal L(V_\infty)$ of computably enumerable vector subspaces of $V_\infty$ and its quotient lattice modulo finite dimension subspaces, $\mathcal L^*(V_\infty)$, have been studied extensively. At the same time, many important questions still remain open. R. Downey and J. Remmel [question 5.8, p. 1031, in: Yu. L. Ershov (ed.) et al., Handbook of recursive mathematics. Vol. 2: Recursive algebra, analysis and combinatorics (Stud. Logic Found. Math., 139), Amsterdam, Elsevier, 1998] posed the question of finding meaningful orbits in $\mathcal L^*(V_\infty)$. We believe that this question is important and difficult and its answer depends on significant progress in the structure theory for the lattice $\mathcal L^*(V_\infty)$, and also on a better understanding of its automorphisms. Here we give a necessary and sufficient condition for quasimaximal (hence maximal) vector spaces with extendable bases to be in the same orbit of $\mathcal L^*(V_\infty)$.
More specifically, we consider two vector spaces, $V_1$ and $V_2$, which are spanned by two quasimaximal subsets of, possibly different, computable bases of $V_\infty$. We give a necessary and sufficient condition for the principal filters determined by $V_1$ and $V_2$ in $\mathcal L^*(V_\infty)$ to be isomorphic. We also specify a necessary and sufficient condition for the existence of an automorphism $\Phi$ of $\mathcal L^*(V_\infty)$ such that $\Phi$ maps the equivalence class of $V_1$ to the equivalence class of $V_2$. Our results are expressed using m-degrees of relevant sets of vectors.
This study parallels the study of orbits of quasimaximal sets in the lattice $\mathcal E$ of computably enumerable sets, as well as in its quotient lattice modulo finite sets, $\mathcal E^*$, carried out by R. Soare in [Ann. Math. (2), 100 (1974), 80–120]. However, our conclusions and proof machinery are quite different from Soare's. In particular, we establish that the structure of the principal filter determined by a quasimaximal vector space in $\mathcal L^*(V_\infty)$ is generally much more complicated than the one of a principal filter determined by a quasimaximal set in $\mathcal E^*$. We also state that, unlike in $\mathcal E^*$, having isomorphic principal filters in $\mathcal L^*(V_\infty)$ is merely a necessary condition for the equivalence classes of two quasimaximal vector spaces to be in the same orbit of $\mathcal L^*(V_\infty)$.