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Local Initial Segments of The Turing Degrees

Published online by Cambridge University Press:  15 January 2014

Bjørn Kjos-Hanssen
Bjørn Kjos-Hanssen*
Affiliation:
Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, D-69120 Heidelberg, Germany.E-mail:[email protected]
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Abstract

Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 9 , Issue 1 , March 2003 , pp. 26 - 36
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Abraham, U. and Shore, R. A., Initial segments of the degrees of size 풩1 , Israel Journal of Mathematics, vol. 53 (1986), no. 1, pp. 1–51.CrossRefGoogle Scholar
[2] Gierz, G. et al, A compendium of continuous lattices, Springer Verlag, 1980.CrossRefGoogle Scholar
[3] Grätzer, G., General lattice Theory, Birkhäuser Verlag, Basel, 1998.Google Scholar
[4] Grätzer, G. and Schmidt, E. T., Characterization of congruence lattices of abstract algebra, Acta Scientiarum Mathematicarum Szeged, vol. 24 (1963), pp. 34–59.Google Scholar
[5] Groszek, M. J. and Slaman, T. A., Independence results on the global structure of the Turing degrees, Transactions of the American Mathematical Society, vol. 277 (1983), pp. 579–587.Google Scholar
[6] Jockusch, C. G. Jr. and Posner, D. B., Double jumps of minimal degrees, The Journal of Symbolic Logic, vol. 43 (1978), no. 4, pp. 715–724.Google Scholar
[7] Jónsson, B., On the representation of lattices, Mathematica Scandinavica, vol. 1 (1953), pp. 193–206.Google Scholar
[8] Kjos-Hanssen, B., Lattice initial segments of the Turing degrees, Ph.D. thesis , University of California at Berkeley, 2002.Google Scholar
[9] Lachlan, A. H., Solution to a problem of Spector, Canadian Journal of Mathematics, vol. 23 (1971), pp. 247–256.Google Scholar
[10]; Lachlan, A. H. and Lebeuf, R., Countable initial segments of the degrees, The Journal of Symbolic Logic, vol. 41 (1976), pp. 289–300.Google Scholar
[11] Lerman, M., Initial segments of the degrees of unsolvability, Annals of Mathematics (2), vol. 93 (1971), pp. 311–389.Google Scholar
[12] Lerman, M., Degrees of unsolvability, Perspectives in Mathematical Logic, Springer-Verlag, Heidelberg, 1983.CrossRefGoogle Scholar
[13] Malcev, A. I., On the general theory of algebraic systems, Mathematics Sbornik, vol. 35 (1954), pp. 3–20.Google Scholar
[14] Malcev, A. I., On the general theory of algebraic systems, Transactions of the American Mathematical Society, vol. 27 (1963), pp. 125–142.Google Scholar
[15] Miller, W. and Martin, D. A., The degrees of hyperimmune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 159–166.Google Scholar
[16] Nies, A., Shore, R. A., and Slaman, T. A., Definability in the recursively enumerable degrees, this Bulletin, vol. 2 (1996), no. 4, pp. 392–404.Google Scholar
[17] Pudlák, P., A new proof of the congruence lattice representation theorem, Algebra Universalis, vol. 6 (1976), pp. 269–275.Google Scholar
[18] Sacks, G. E., A minimal degree below 0′, Bulletin of the American Mathematical Society, vol. 67 (1961), pp. 416–419.Google Scholar
[19] Shore, R.A., Defining jump classes in the degrees below 0′, Proceedings of the American Mathematical Society, vol. 104 (1988), no. 1, pp. 287–292.Google Scholar
[20] Shore, R. A. and Slaman, T. A., Defining the Turing jump, Mathematical Research Letters, vol. 6 (1999), no. 5–6, pp. 711–722.Google Scholar
[21] Slaman, T. A. and Woodin, W. H., Definability in degree structures, to appear.Google Scholar
[22] Spector, C., On the degrees of recursive unsolvability, Annals of Mathematics (2), vol. 64 (1956), pp. 581–592.CrossRefGoogle Scholar
[23] Whitman, P.M., Lattices, equivalence relations and subgroups, Bulletin of the American Mathematical Society, vol. 52 (1946), pp. 507–522.Google Scholar