Orbits of Maximal Vector Spaces

Abstract

Let V_∞be a standard computable infinite-dimensional vector space over the field of rationals. The lattice\( \mathfrak{L} \)(V_∞) of computably enumerable vector subspaces of V_∞and its quotient lattice modulo finite dimension,\( \mathfrak{L} \)^*(V_∞), have been studied extensively. At the same time, many important questions still remain open. In 1998, R. Downey and J. Remmel posed the question of finding meaningful orbits in\( \mathfrak{L} \)^*(V_∞) [4, Question 5.8]. This question is important and difficult and its answer depends on significant progress in the structure theory for the lattice\( \mathfrak{L} \)^*(V_∞), 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\( \mathfrak{L} \)^*(V_∞). More specifically, we consider two vector spaces, V₁and V₂, which are spanned by two quasimaximal subsets of, possibly different, computable bases of V_∞. We give a necessary and sufficient condition for the principal filters determined by V₁and V₂in\( \mathfrak{L} \)^*(V_∞) to be isomorphic. We also specify a necessary and sufficient condition for the existence of an automorphism Φ of\( \mathfrak{L} \)^*(V_∞) such that Φ maps the equivalence class of V₁to the equivalence class of V₂. 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 ε of computably enumerable sets, as well as in its quotient lattice modulo finite sets, ε^*, carried out by R. Soare in [13]. 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\( \mathfrak{L} \)^*(V_∞) is generally much more complicated than the one of a principal filter determined by a quasimaximal set in ε^*. We also state that, unlike in ε^*, having isomorphic principal filters in\( \mathfrak{L} \)^*(V_∞) is merely a necessary condition for the equivalence classes of two quasimaximal vector spaces to be in the same orbit of\( \mathfrak{L} \)^*(V_∞).