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6 papers
Orbits of maximal vector spaces
R. D. Dimitrova,
V. Harizanovb a Department of Mathematics, Western Illinois University, Macomb, IL, 61455, USA
b Department of Mathematics, George Washington University, Washington, DC, 20052, USA
Abstract:
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
$\mathcal L^*(V_\infty)$.
Keywords:
infinite-dimensional vector space over field of rationals, quasimaximal set, equivalence classes, principal filter, orbit, lattice.
UDC:
510.5 Received: 09.07.2014
DOI:
10.17377/alglog.2015.54.603
Fulltext:
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