The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. Here we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree d, we investigate the groups of d-computable automorphisms of the lattice of d-computably enumerable vector spaces, of the interval Boolean algebra Bη of the ordered set of rationals, and of the lattice of d-computably enumerable subalgebras of Bη. For these groups, we show that Turing reducibility can be used to substitute the group-theoretic embedding. We also prove that the Turing degree of the isomorphism types for these groups is the second Turing jump d′′ of d.
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References
B. L. Van der Waerden, Algebra, Vols. I, II, Springer, New York (2003).
R. I. Soare, Recursively Enumerable Sets and Degrees, Persp. Math. Log., Omega Ser., Springer, Berlin (1987).
H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967).
S. S. Goncharov and Yu. L. Ershov, Constructive Models, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).
E. B. Fokina, V. Harizanov, and A. Melnikov, “Computable model theory,” in Turing’s Legacy: Developments from Turing’s Ideas in Logic, Lect. Notes Log., 42, R. Downey (ed.), Cambridge Univ. Press, Ass. Symb. Log., Cambridge (2014), pp. 124-194.
A. S. Morozov, “Groups of computable automorphisms, in Handbook of Recursive Mathematics, Stud. Log. Found. Math., 138, Y. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel (eds.), Elsevier, Amsterdam (1998), pp. 311-345.
A. S. Morozov, “Permutations and implicit definability,” Algebra and Logic, 27, No. 1, 12-24 (1988).
A. S. Morozov, “Turing reducibility as algebraic embeddability,” Sib. Math. J., 38, No. 2, 312-313 (1997).
A. S. Morozov, “On theories of classes of groups of recursive permutations,” in Mathematical Logic and Algorithm Theory, Trudy Inst. Mat. SO AN SSSR [in Russian], 12 (1989), pp. 91-104.
A. S. Morozov, “Computable groups of automorphisms of models,” Algebra and Logic, 25, No. 4, 261-266 (1986).
S. S. Goncharov, V. Harizanov, J. Knight, A. S. Morozov, and A. V. Romina, “On automorphic tuples of elements in computable models,” Sib. Math. J., 46, No. 3, 405-412 (2005).
J. F. Knight, “Degrees coded in jumps of orderings,” J. Symb. Log., 51, No. 4, 1034-1042 (1986).
V. Harizanov and R. Miller, “Spectra of structures and relations,” J. Symb. Log., 72, No. 1, 324-348 (2007).
L. J. Richter, Degrees of unsolvability of models, Ph.D. Thesis, Univ. Illinois at Urbana- Champaign (1977).
L. Richter, “Degrees of structures,” J. Symb. Log., 46, No. 4, 723-731 (1981).
R. Dimitrov, V. Harizanov, and A. Morozov, “Automorphism groups of substructure lattices of vector spaces in computable algebra,” in Lect. Notes Comput. Sci., 9709, Springer, Cham (2016), pp. 251-260.
G. Metakides and A. Nerode, “Recursively enumerable vector spaces,” Ann. Math. Log., 11, 147-171 (1977).
R. G. Downey and J. B. Remmel, “Computable algebras and closure systems: Coding properties,” in Handbook of Recursive Mathematics, Stud. Log. Found. Math., 139, Y. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel (eds.), Elsevier, Amsterdam (1998), pp. 997-1039.
R. Dimitrov, V. Harizanov, and A. Morozov, “Dependence relations in computably rigid computable vector spaces,” Ann. Pure Appl. Log., 132, No. 1, 97-108 (2005).
R. G. Downey, D. R. Hirschfeldt, A. M. Kach, S. Lempp, J. R. Mileti, and A. Montalbán, “Subspaces of computable vector spaces,” J. Alg., 314, No. 2, 888-894 (2007).
D. R. Guichard, “Automorphisms of substructure lattices in recursive algebra,” Ann. Pure Appl. Log., 25, 47-58 (1983).
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Supported by the National Science Foundation, binational research grant DMS-1101123.
Supported by the Simons Foundation Collaboration Grant and by CCFF and Dean’s Research Chair awards of the George Washington University.
Translated from Algebra i Logika, Vol. 59, No. 1, pp. 27-47, January-February, 2020.
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Dimitrov, R.D., Harizanov, V. & Morozov, A.S. Turing Degrees and Automorphism Groups of Substructure Lattices. Algebra Logic 59, 18–32 (2020). https://doi.org/10.1007/s10469-020-09576-x
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DOI: https://doi.org/10.1007/s10469-020-09576-x