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Degree theoretical splitting properties of recursively enumerable sets

Published online by Cambridge University Press:  12 March 2014

Klaus Ambos-Spies  and
Peter A. Fejer
Klaus Ambos-Spies*
Affiliation:
Lehrstuhl Informatik II, Universität Dortmund, 4600 Dortmund 50, Bundesrepublik Deutschlands
Peter A. Fejer
Affiliation:
Department of Mathematics and Computer Science, University of Massachusetts at Boston, Boston, Massachusetts 02125
*
Fachbereich Informatik, Universität Oldenburg, D-2900 Oldenburg, West Germany
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A recursively enumerable splitting of an r.e. set A is a pair of r.e. sets B and C such that A = B ∪ C and B ∩ C = ⊘. Since for such a splitting deg A = deg B ∪ deg C, r.e. splittings proved to be a quite useful notion for investigations into the structure of the r.e. degrees. Important splitting theorems, like Sacks splitting [S1], Robinson splitting [R1] and Lachlan splitting [L3], use r.e. splittings.

Since each r.e. splitting of a set induces a splitting of its degree, it is natural to study the relation between the degrees of r.e. splittings and the degree splittings of a set. We say a set A has the strong universal splitting property (SUSP) if each splitting of its degree is represented by an r.e. splitting of itself, i.e., if for deg A = b ∪ c there is an r.e. splitting B, C of A such that deg B = b and deg C = c. The goal of this paper is the study of this splitting property.

In the literature some weaker splitting properties have been studied as well as splitting properties which imply failure of the SUSP.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 4 , December 1988 , pp. 1110 - 1137
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[A1] Ambos-Spies, K., On the structure of the recursively enumerable degrees, Dissertation, Universität München, Munich, 1980.Google Scholar
[A2] Ambos-Spies, K., Anti-mitotic recursively enumerable sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 31 (1985), pp. 461–477.CrossRefGoogle Scholar
[A3] Ambos-Spies, K., Contiguous r.e. degrees, Computation and proof theory (Logic Colloquium '83, Aachen, part II; Richter, M. et al., editors), Lecture Notes in Mathematics, vol. 1104, Springer-Verlag, Berlin, 1984, pp. 1–37.CrossRefGoogle Scholar
[A4] Ambos-Spies, K., Cupping and noncapping in the r.e. wtt and Turing degrees, Archive for Mathematical Logic (to appear).Google Scholar
[C1] Cohen, P. F., Weak truth table reducibility and the pointwise ordering of 1-1 recursive functions, Ph.D. thesis, University of Illinois, Urbana, Illinois, 1975.Google Scholar
[D1] Downey, R. G., The degrees of r.e. sets without the universal splitting property, Transactions of the American Mathematical Society, vol. 291 (1985), pp. 337–351.CrossRefGoogle Scholar
[F1] Fejer, P. A., Branching degrees above low degrees, Transactions of the American Mathematical Society, vol. 273 (1982), pp. 157–180.CrossRefGoogle Scholar
[F2] Fejer, P. A., The density of the nonbranching degrees, Annals of Pure and Applied Logic, vol. 24 (1983), pp. 113–130.CrossRefGoogle Scholar
[FS] Fejer, P. A. and Soare, R. I., The plus-cupping theorem for the recursively enumerable degrees, Logic year 1979–80(Lerman, M. et al., editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, 1980, pp. 49–62.CrossRefGoogle Scholar
[I1] Ingrassia, M. A., P-genericity for recursively enumerable sets, Ph.D. thesis, University of Illinois, Urbana, Illinois, 1981.Google Scholar
[L1] Lachlan, A. H., The priority method. I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematick, vol. 13 (1967), pp. 1–10.CrossRefGoogle Scholar
[L2] Lachlan, A. H., Embedding nondistributive lattices in the recursively enumerable degrees, Conference in mathematical logic—London '70 (Hodges, W., editor), Lecture Notes in Mathematics, vol. 255, Springer-Verlag, Berlin, 1972, pp. 149–177.CrossRefGoogle Scholar
[L3] Lachlan, A. H., Decomposition of recursively enumerable degrees, Proceedings of the American Mathematical Society, vol. 79 (1980), pp. 629–634.CrossRefGoogle Scholar
[L4] Ladner, R. E., Mitotic recursively enumerable sets, this Journal, vol. 38 (1973), pp. 199–211.Google Scholar
[L5] Ladner, R. E., A completely mitotic nonrecursive r.e. degree, Transactions of the American Mathematical Society, vol. 184 (1973), pp. 479–507.CrossRefGoogle Scholar
[LS] Ladner, R. E. and Sasso, L.P. Jr., The weak truth table degrees of recursively enumerable sets, Annals of Mathematical Logic, vol. 8 (1975), pp. 429–448.CrossRefGoogle Scholar
[LR1] Lerman, M. and Remmel, J. B., The universal splitting property. I, Logic Colloquium '80 (Dalen, D. van et al., editors), Studies in Logic and the Foundations of Mathematics, vol. 108, North-Holland, Amsterdam, 1982, pp. 181–207.Google Scholar
[LR2] Lerman, M. and Remmel, J. B., The universal splitting property. II, this Journal, vol. 49 (1984), pp. 137–150.Google Scholar
[R1] Robinson, R. M., Interpolation and embedding in the recursively enumerable degrees, Annals of Mathematics, ser. 2, vol. 93 (1971), pp. 285–314.CrossRefGoogle Scholar
[R2] Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[S1] Sacks, G. E., Degrees of unsolvability, Annals of Mathematics Studies, vol. 55, Princeton University Press, Princeton, New Jersey, 1966.Google Scholar
[S2] Slaman, T., The density of the branching degrees (to appear).Google Scholar
[S3] Soare, R. I., Computational complexity, speedable and levelable sets, this Journal, vol. 42 (1977), pp. 545–563.Google Scholar
[S4] Soare, R. I., Fundamental methods for constructing recursively enumerable degrees, Recursion theory: its generalisations and applications (proceedings of Logic Colloquium '79, Leeds; Drake, F. R. and Wainer, S. S., editors), London Mathematical Society Lecture Note Series, vol. 45, Cambridge University Press, Cambridge, 1980, pp. 1–51.Google Scholar
[S5] Stob, M., wtt- and T-degrees of r.e. sets, this Journal, vol. 48 (1983), pp. 921–930.Google Scholar
[Y1] Yates, C. E. M., On the degrees of index sets. II, Transactions of the American Mathematical Society, vol. 135 (1969), pp. 249–266.CrossRefGoogle Scholar