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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Csima, B.F., Franklin, J.N.Y., Shore, R.A.: Degrees of categoricity and the hyperarithmetic hierarchy (submitted)
Dummit D.S., Foote R.M.: Abstract Algebra. 3rd edn. Wiley, Hoboken (2004)
Fokina E.B., Kalimullin I., Miller R.G.: Degrees of categoricity of computable structures. Arch. Math. Logic 49, 51–67 (2010)
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)
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)
Harizanov V.S., Miller R.G.: Spectra of structures and relations. J. Symb. Logic 72, 324–348 (2007)
Hirschfeldt, D.R., Kramer, K., Miller, R.G., Shlapentokh, A.: Categoricity properties for computable algebraic fields (submitted)
Hodges W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)
Jockusch C.G., Soare R.I.: Π0 1-classes and degrees of theories. Trans. Am. Math. Soc. 173, 33–56 (1972)
Knight J.F.: Degrees coded in jumps of orderings. J. Symb. Logic 51, 1034–1042 (1986)
Miller, R.G.: Computability, Definability, Categoricity, and Automorphisms. Ph.D. dissertation
Miller R.G.: d-Computable categoricity for algebraic fields. J. Symb. Logic 74, 1325–1351 (2009)
Miller R.G.: Computable fields and Galois theory. Notices Am. Math. Soc. 55, 798–807 (2008)
Miller, R.G., Shlapentokh, A.: Computable categoricity for algebraic fields with splitting algorithms (submitted)
Rothmaler, P.: Introduction to Model Theory. Algebra, Logic and Applications, vol. 15. Gordon and Breach (2000)
Soare R.I.: Recursively Enumerable Sets and Degrees. Springer, Berlin (1987)
Soare, R.I.: Computability Theory and Applications: The Art of Classical Computability. two volumes. Springer, Heidelberg (to appear)
Steiner, R.M.: Reducibility, Degree Spectra, and Lowness in Algebraic Structures. Ph.D. thesis
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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