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Automorphism Groups of Substructure Lattices of Vector Spaces in Computable Algebra

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Pursuit of the Universal (CiE 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9709))

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Abstract

For a Turing degree \(\mathbf {x}\), we investigate the automorphisms of the lattice of \(\mathbf {x}\)-c.e. vector spaces. We establish the equivalence of the embedding relation for these automorphism groups with the order relation on the corresponding Turing degrees. By a result of Guichard the automorphisms of the lattice of \(\mathbf {x}\)-c.e. vector spaces are induced by \(\mathbf {x}\)-computable invertible semilinear transformations, GSL\(_{\mathbf {x}}\). We prove that the Turing degree spectrum of the group GSL\(_{\mathbf {x}}\) is the upper cone of Turing degrees \(\ge \mathbf {x}^{\prime \prime }\).

The three authors acknowledge partial support of the binational research grant DMS-1101123 from the National Science Foundation. The second author acknowledges support of the NSF grant DMS-1202328 and the Columbian College of Arts and Sciences of GWU.

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Correspondence to Rumen Dimitrov .

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Dimitrov, R., Harizanov, V., Morozov, A. (2016). Automorphism Groups of Substructure Lattices of Vector Spaces in Computable Algebra. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_26

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  • DOI: https://doi.org/10.1007/978-3-319-40189-8_26

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