Skip to main content
Log in

Turing Degrees and Automorphism Groups of Substructure Lattices

  • Published:
Algebra and Logic Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. B. L. Van der Waerden, Algebra, Vols. I, II, Springer, New York (2003).

  2. R. I. Soare, Recursively Enumerable Sets and Degrees, Persp. Math. Log., Omega Ser., Springer, Berlin (1987).

  3. H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967).

    MATH  Google Scholar 

  4. S. S. Goncharov and Yu. L. Ershov, Constructive Models, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).

  5. 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.

  6. 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.

  7. A. S. Morozov, “Permutations and implicit definability,” Algebra and Logic, 27, No. 1, 12-24 (1988).

  8. A. S. Morozov, “Turing reducibility as algebraic embeddability,” Sib. Math. J., 38, No. 2, 312-313 (1997).

    Article  MathSciNet  Google Scholar 

  9. 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.

  10. A. S. Morozov, “Computable groups of automorphisms of models,” Algebra and Logic, 25, No. 4, 261-266 (1986).

  11. 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).

    Article  MathSciNet  Google Scholar 

  12. J. F. Knight, “Degrees coded in jumps of orderings,” J. Symb. Log., 51, No. 4, 1034-1042 (1986).

    Article  MathSciNet  Google Scholar 

  13. V. Harizanov and R. Miller, “Spectra of structures and relations,” J. Symb. Log., 72, No. 1, 324-348 (2007).

    Article  MathSciNet  Google Scholar 

  14. L. J. Richter, Degrees of unsolvability of models, Ph.D. Thesis, Univ. Illinois at Urbana- Champaign (1977).

  15. L. Richter, “Degrees of structures,” J. Symb. Log., 46, No. 4, 723-731 (1981).

    Article  MathSciNet  Google Scholar 

  16. 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.

  17. G. Metakides and A. Nerode, “Recursively enumerable vector spaces,” Ann. Math. Log., 11, 147-171 (1977).

    Article  MathSciNet  Google Scholar 

  18. 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.

  19. 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).

    Article  MathSciNet  Google Scholar 

  20. 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).

  21. D. R. Guichard, “Automorphisms of substructure lattices in recursive algebra,” Ann. Pure Appl. Log., 25, 47-58 (1983).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. D. Dimitrov.

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10469-020-09576-x

Keywords

Navigation