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Elementary theories and structural properties of d-c.e. and n-c.e. degrees

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Abstract

This paper is a survey on the upper semilattices of Turing and enumeration degrees of n-c.e. sets. Questions on the structural properties of these semilattices, and some model-theoretic properties are considered.

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References

  1. M. M. Arslanov, “Structural properties of the degrees below 0’,” Sov. Math. Dokl. N.S. 283, 270–273 (1985).

    MathSciNet  Google Scholar 

  2. M. M. Arslanov, “On the upper semilattice of Turing degrees below 0’,” Russian Mathematics (Iz. VUZ) 32 (7), 27–33 (1988).

    MathSciNet  MATH  Google Scholar 

  3. M. M. Arslanov, S. B. Cooper, and A. Li, “There is no low maximal d.c.e. degree,” Math. Logic Quart. 46, 409–416 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. M. Arslanov, S. B. Cooper, and A. Li, “There is no low maximal d.c.e. degree—Corrigendum,” Math. Logic Quart. 50, 628–636 (2004).

    Article  MathSciNet  Google Scholar 

  5. M. M. Arslanov, I. Sh. Kalimullin, and A. Sorbi, “Density results in the d0 2 e-degrees,” Archive for Mathematical Logic 40, 597–614 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  6. M. M. Arslanov, I. Sh. Kalimullin, and S. Lempp, “On Downey’s conjecture,” J. Symb. Logic 75, 401–441 (2010).

    Article  MATH  Google Scholar 

  7. M. M. Arslanov, G. LaForte, and T. A. Slaman, “Relative enumerability in the difference hierarchy,” J. Symb. Logic 63, 411–420 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. M. Arslanov, S. Lempp, and R. A. Shore, “Interpolating d-r.e. and REA degrees between r.e. degrees,” Annals of Pure and Applied Logic 78, 29–56 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  9. S. B. Cooper, Degrees of Unsolvability, Ph. D. Thesis (Leicester University, Leicester, England, 1971).

    Google Scholar 

  10. S. B. Cooper, “The density of the Low2 n-r.e. degrees,” Archive for Math. Logic 31, 19–24 (1991)

    Article  MATH  Google Scholar 

  11. S. B. Cooper, “A splitting theorem for the n-r.e. degrees,” Proc. Amer. Math. Soc. 115, 461–471 (1992).

    MathSciNet  MATH  Google Scholar 

  12. S. B. Cooper, L. Harrington, A. H. Lachlan, S. Lempp, and R. I. Soare, “The d-r.e. degrees are not dense,” Ann. Pure and Applied Logic 55, 125–151 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  13. S. B. Cooper and A. Li, “Non-uniformity and generalised Sacks splitting,” Acta Mathematica Sinica, English Series 18, 327–334 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. B. Cooper and A. Li, “Splitting and cone avoidance in the d.c.e. degrees,” Science in China (Series A) 45, 1135–1146 (2002).

    MathSciNet  MATH  Google Scholar 

  15. S. B. Cooper and X. Yi, “Isolated d.r.e. degrees,” Preprint series 17 (Universiry of Leeds, Dept of Pure Math., 1995).

    Google Scholar 

  16. R. G. Downey, “D.r.e. degrees and the nondiamond theorem,” Bull. London Math. Soc. 21, 43–50 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  17. R. G. Downey and L. Yu, “There are no maximal low D.C.E. Degrees,” Notre Dame J. Formal Logic 45 (3), 147–159 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. L. Ershov, “On a hierarchy of sets I,” Algebra i Logika 7 (1), 47–73 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  19. Y. L. Ershov, “On a hierarchy of sets II,” Algebra i Logika 7 (4), 15–47 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. L. Ershov, “On a hierarchy of sets III,” Algebra i Logika 9 (1), 34–51 (1970).

    Article  MathSciNet  Google Scholar 

  21. M. Kh. Faizrakhmanov, “Decomposability of low 2-computably enumerable degrees and Turing jumps in the Ershov hierarchy,” Russian Mathematics (Iz. VUZ) 54 (12), 51–58 (2010).

    MathSciNet  MATH  Google Scholar 

  22. Fang, G. Wu, and M. M. Yamaleev, “On a problem of Ishmukhametov,” Arch. Math. Logic 52, 733–741 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  23. Ch. Fang, J. Liu, G. Wu, and M. M. Yamaleev, Nonexistence of minimal pairs in L[d], Evolving Computability (Proceeding of CiE 2015). The series Lecture Notes in Computer Science 9136, 177–185 (2015).

    MathSciNet  Google Scholar 

  24. J. Liu, G. Wu, and M. M. Yamaleev, “Downward density of exact degrees,” Lobachevskii Journal of Mathematics 36 (4), 389–398 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  25. H. A. Ganchev and M. I. Soskova, “Definability via Kalimullin pairs in the structure of the enumeration degrees,” Transactions of the AmericanMathematical Society 367, 4873–4893 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  26. E.M. Gold, “Limiting recursion,” J. Symb. Logic 30, 28–48 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  27. Sh. T. Ishmukhametov, “On the r.e. predecessors of d.r.e. degrees,” Arch.Math. Logic 38, 373–386 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  28. Sh. T. Ishmukhametov, “On relative enumerability of Turing degrees,” Arch. Math. Logic 39, 145–154 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  29. C. Jockusch and R. Shore, “Pseudojump operators II: transfinite iterations, hierarchies and minimal covers,” Journal of Symbolic Logic 49, 1205–1236 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  30. I. Sh. Kalimullin, “Elementary theories of semilattices of n-recursive enumerable degrees with respect to enumerability,” Russian Mathematics (Iz. VUZ) 45 (4), 24–27 (2001).

    MathSciNet  MATH  Google Scholar 

  31. I. Sh. Kalimullin, “Splitting properties of n-c.e. enumeration degrees,” Journal of Symbolic Logic 67, 537–546 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  32. I. Sh. Kalimullin, “Definability of the jump operator in the enumeration degrees,” J. Mathematical Logic 3 (2), 257–267 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  33. I. Sh. Kalimullin, “Elementary differences between the (2p)-c.e. and the (2p+1)-c.e. enumeration degrees,” J. Symbolic Logic 72, 277–284 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  34. A. H. Lachlan, “Lower bounds for pairs of recursively enumerable degrees,” Proc. London Math. Soc. 16, 537–569 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  35. H. Putnam, “Trial and error predicates and the solution to a problem of Mostowski,” J. Symb. Logic 30, 49–57 (1965).

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  37. L. Welch, A Hierarchy of Families of Recursively Enumerable Degrees and a Theorem of Founding Minimal Pairs, Ph. D. Thesis (University of Illinois, Urbana, USA, 1980).

    Google Scholar 

  38. G. Wu, “Isolation and lattice embedding,” J. Symb. Logic 67, 1055–1064 (2002).

    Article  MATH  Google Scholar 

  39. G. Wu and M. M. Yamaleev, “Isolation: motivations and applications,” Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki 154 (2), 204–217 (2012).

    MathSciNet  Google Scholar 

  40. M. M. Yamaleev, “The splitting of 2-computably enumerable degrees avoiding the upper cones,” Russian Mathematics (Iz. VUZ) 53 (6), 76–80 (2009).

    MathSciNet  MATH  Google Scholar 

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Correspondence to M. M. Arslanov.

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Submitted by F. M. Ablayev

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Arslanov, M.M., Kalimullin, I.S. & Yamaleev, M.M. Elementary theories and structural properties of d-c.e. and n-c.e. degrees. Lobachevskii J Math 37, 713–722 (2016). https://doi.org/10.1134/S199508021606010X

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