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, V 1 and V 2 , 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 V 1 and V 2 in \( \mathfrak{L} \) *(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 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 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 ∞ ).
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(V. Harizanov) Supported by the NSF (grant DMS-1202328) and by the GWU Columbian College Facilitating Fund.
Translated from Algebra i Logika, Vol. 54, No. 6, pp. 680–732, November-December, 2015.
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Dimitrov, R.D., Harizanov, V. Orbits of Maximal Vector Spaces. Algebra Logic 54, 440–477 (2016). https://doi.org/10.1007/s10469-016-9366-9
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DOI: https://doi.org/10.1007/s10469-016-9366-9