Abstract
We survey fundamental notions and results in the study of the lattice of computably enumerable vector spaces and its quotient lattice modulo finite dimension. These lattices were introduced and first studied by Metakides and Nerode in the late 1970s and later extensively investigated by Downey, Remmel and others. First, we focus on the role of the dependence algorithm, the effectiveness of the bases, and the Turing degree-theoretic complexity of the dependence relations. We present a result on the undecidability of the theories of the above lattices. We show the development of various notions of maximality for vector spaces, and role they play in the study of lattice automorphisms and automorphism bases. We establish a new result about the role of supermaximal spaces in the quotient lattice automorphism bases. Finally, we discuss the problem of finding orbits of maximal spaces and the recent progress on this topic.
Harizanov was partially supported by the NSF grant 1202328, by the CCFF award and CCAS Dean’s research chair award of the George Washington University, as well as by the Simons Foundation Collaboration Grants for Mathematicians.
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Dimitrov, R.D., Harizanov, V. (2017). The Lattice of Computably Enumerable Vector Spaces. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_23
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