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Effective algebraicity

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Abstract

Results of R. Miller in 2009 proved several theorems about algebraic fields and computable categoricity. Also in 2009, A. Frolov, I. Kalimullin, and R. Miller proved some results about the degree spectrum of an algebraic field when viewed as a subfield of its algebraic closure. Here, we show that the same computable categoricity results also hold for finite-branching trees under the predecessor function and for connected, finite-valence, pointed graphs, and we show that the degree spectrum results do not hold for these trees and graphs. We also offer an explanation for why the degree spectrum results distinguish these classes of structures: although all three structures are algebraic structures, the fields are what we call effectively algebraic.

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References

  1. Csima, B.F., Franklin, J.N.Y., Shore, R.A.: Degrees of categoricity and the hyperarithmetic hierarchy (submitted)

  2. Dummit D.S., Foote R.M.: Abstract Algebra. 3rd edn. Wiley, Hoboken (2004)

    MATH  Google Scholar 

  3. Fokina E.B., Kalimullin I., Miller R.G.: Degrees of categoricity of computable structures. Arch. Math. Logic 49, 51–67 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Frolov, A., Kalimullin, I., Miller, R.G. Spectra of algebraic fields and subfields, mathematical theory and computational practice. In: Fifth Conference on Computability in Europe (Lecture Notes in Computer Science 5635), pp. 232–241. Springer, Berlin (2009)

  5. Harizanov, V.S.: Pure Computable Model Theory, handbook of Recursive Mathematics Vol. 1 Studies in Logic and the Foundations of Mathematics, vol. 138, pp. 3–114 (1998)

  6. Harizanov V.S., Miller R.G.: Spectra of structures and relations. J. Symb. Logic 72, 324–348 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hirschfeldt, D.R., Kramer, K., Miller, R.G., Shlapentokh, A.: Categoricity properties for computable algebraic fields (submitted)

  8. Hodges W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  9. Jockusch C.G., Soare R.I.: Π0 1-classes and degrees of theories. Trans. Am. Math. Soc. 173, 33–56 (1972)

    MathSciNet  MATH  Google Scholar 

  10. Knight J.F.: Degrees coded in jumps of orderings. J. Symb. Logic 51, 1034–1042 (1986)

    Article  MATH  Google Scholar 

  11. Miller, R.G.: Computability, Definability, Categoricity, and Automorphisms. Ph.D. dissertation

  12. Miller R.G.: d-Computable categoricity for algebraic fields. J. Symb. Logic 74, 1325–1351 (2009)

    Article  MATH  Google Scholar 

  13. Miller R.G.: Computable fields and Galois theory. Notices Am. Math. Soc. 55, 798–807 (2008)

    MATH  Google Scholar 

  14. Miller, R.G., Shlapentokh, A.: Computable categoricity for algebraic fields with splitting algorithms (submitted)

  15. Rothmaler, P.: Introduction to Model Theory. Algebra, Logic and Applications, vol. 15. Gordon and Breach (2000)

  16. Soare R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)

    Google Scholar 

  17. Soare, R.I.: Computability Theory and Applications: The Art of Classical Computability. two volumes. Springer, Heidelberg (to appear)

  18. Steiner, R.M.: Reducibility, Degree Spectra, and Lowness in Algebraic Structures. Ph.D. thesis

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Correspondence to Rebecca M. Steiner.

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The author was partially supported by grant #DMS-1001306 from the National Science Foundation.

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Steiner, R.M. Effective algebraicity. Arch. Math. Logic 52, 91–112 (2013). https://doi.org/10.1007/s00153-012-0308-5

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  • DOI: https://doi.org/10.1007/s00153-012-0308-5

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