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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Dimitrov, R., Harizanov, V., Morozov, A.S.: Dependence relations in computably rigid computable vector spaces. Ann. Pure Appl. Logic 132, 97–108 (2005)
Downey, R.G., Remmel, J.B.: Computable algebras, closure systems: coding properties. In: Ershov, Y., Goncharov, S.S., Nerode, A., Remmel, J.B. (eds.) Handbook of Recursive Mathematics, vol. 2. Studies in Logic and the Foundations of Mathematics, vol. 139, pp. 997–1039. North-Holland, Amsterdam (1998)
Ershov, Y.L., Goncharov, S.S.: Constructive Models. Siberian School of Algebra and Logic. Kluwer Academic/Plenum Publishers, New York (2000). (English translation)
Fokina, E., Harizanov, V., Melnikov, A.: Computable model theory. In: Downey, R. (ed.) Turing’s Legacy: Developments from Turing Ideas in Logic, pp. 124–194. Cambridge University Press/ASL, Cambridge (2014)
Goncharov, S., Harizanov, V., Knight, J., Morozov, A., Romina, A.: On automorphic tuples of elements in computable models. Siberian Math. J. 46, 405–412 (2005). (English translation)
Guichard, D.R.: Automorphisms of substructure lattices in recursive algebra. Ann. Pure Appl. Logic 25, 47–58 (1983)
Harizanov, V., Miller, R.: Spectra of structures and relations. J. Sym. Logic 72, 324–348 (2007)
Knight, J.F.: Degrees coded in jumps of orderings. J. Sym. Logic 51, 1034–1042 (1986)
Metakides, G., Nerode, A.: Recursively enumerable vector spaces. Ann. Pure Appl. Logic 11, 147–171 (1977)
Morozov, A.S.: Permutations and implicit definability. Algebra Logic 27, 12–24 (1988). (English translation)
Morozov, A.S.: Turing reducibility as algebraic embeddability. Siberian Math. J. 38, 312–313 (1997). (English translation)
Morozov, A.S.: Groups of computable automorphisms. In: Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B. (eds.) Handbook of Recursive Mathematics, vol. 1. Studies in Logic and the Foundations of Mathematics, vol. 139, pp. 311–345. North-Holland, Amsterdam (1998)
Morozov, A.S., On theories of classes of groups of recursive permutations. Tr. Inst. Matematiki (Novosibirsk) 12 (1989). Mat. Logika i Algoritm. Probl. 91–104 (Russian). (English translation. Siberian Adv. Math. 1, 138–153 (1991))
Morozov, A.S.: Computable groups of automorphisms of models. Algebra Logic 25, 261–266 (1986). (English translation)
Richter, L.J.: Degrees of unsolvability of models. Ph.D. dissertation, University of Illinois at Urbana-Champaign (1977)
Richter, L.J.: Degrees of structures. J. Sym. Logic 46, 723–731 (1981)
Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)
Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)
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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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