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Computability and Complexity pp 366–393Cite as

The Lattice of Computably Enumerable Vector Spaces

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10010))

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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Author information

Authors and Affiliations

  1. Department of Mathematics, Western Illinois University, Macomb, Illinois, 61455, USA

    Rumen D. Dimitrov

  2. Department of Mathematics, George Washington University, Washington, DC, 20052, USA

    Valentina Harizanov

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  1. Rumen D. Dimitrov
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  2. Valentina Harizanov
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Correspondence to Valentina Harizanov .

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Editors and Affiliations

  1. Victoria University of Wellington, Wellington, New Zealand

    Adam Day

  2. University of Bergen, Bergen, Norway

    Michael Fellows

  3. Victoria University of Wellington, Wellington, New Zealand

    Noam Greenberg

  4. University of Auckland, Auckland, New Zealand

    Bakhadyr Khoussainov

  5. Massey University, Auckland, New Zealand

    Alexander Melnikov

  6. University of Bergen, Bergen, Norway

    Frances Rosamond

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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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  • DOI: https://doi.org/10.1007/978-3-319-50062-1_23

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