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The universal splitting property. II

Published online by Cambridge University Press:  12 March 2014

M. Lerman  and
J. B. Remmel
M. Lerman
Affiliation:
University of Connecticut, Storrs, Connecticut 06068
J. B. Remmel
Affiliation:
University of California at San Diego, La Jolla, California 92093
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Extract

We say that a pair of r.e. sets B and C split an r.e. set A if B ∩ C = ∅ and B ∪ C = A. Friedberg [F] was the first to study the degrees of splittings of r.e. sets. He showed that every nonrecursive r.e. set A has a splitting into nonrecursive sets. Generalizations and strengthenings of Friedberg's result were obtained by Sacks [Sa2], Owings [O], and Morley and Soare [MS].

The question which motivated both [LR] and this paper is the determination of possible degrees of splittings of A. It is easy to see that if B and C split A, then both B and C are Turing reducible to A (written B ≤TA and C ≤TA). The Sacks splitting theorem [Sa2] is a result in this direction, as are results by Lachlan and Ladner on mitotic and nonmitotic sets. Call an r.e. set A mitotic if there is a splitting B and C of A such that both B and C have the same Turing degree as A; A is nonmitotic otherwise. Lachlan [Lac] showed that nonmitotic sets exist, and Ladner [Lad1], [Lad2] carried out an exhaustive study of the degrees of mitotic sets.

The Sacks splitting theorem [Sa2] shows that if A is r.e. and nonrecursive, then there are r.e. sets B and C splitting A such that B <TA and C <TA. Since B is r.e. and nonrecursive, we can now split B and continue in this manner to produce infinitely many r.e. degrees below the degree of A which are degrees of sets forming part of a splitting of A. We say that an r.e. set A has the universal splitting property (USP) if for any r.e. set D ≤T A, there is a splitting B and C of A such that B and D are Turing equivalent (written B ≡TD).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 1 , March 1984 , pp. 137 - 150
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

[F]Friedberg, R. M., Three theorems on recursive enumeration, this Journal, vol. 23 (1958), pp. 305–316.Google Scholar
[Lac]Lachlan, A. H., The priority method. I, Zeitschrift fur Mathematische Logik und Grundlager d, Mathematik, vol. 13 (1967), pp. 1–10.CrossRefGoogle Scholar
[Lad1]Ladner, R. E., Mitotic recursively enumerable sets, this Journal, vol. 38 (1973), pp. 199–21Google Scholar
[Lad2]Ladner, R. E., A completely mitotic nonrecursive r.e. degree, Transactions of the America Mathematical Society, vol. 184 (1973), pp. 479–507.CrossRefGoogle Scholar
[LR]Lerman, M. and Remmel, J. B., The universal splitting property. I, Logic Colloquium '80 (va Dalen, D., Editor), North-Holland, Amsterdam, 1982, pp. 181–208.Google Scholar
[MS]Morley, M. D. and Soare, R. I., Boolean algebras, splitting theorems, and sets, Fundament Mathematicae, vol. 90 (1975), pp. 45–52.CrossRefGoogle Scholar
[O]Owings, J. C., Recursion, metarecursion and inclusion, this Journal, vol. 32 (1967), pp. 173–173Google Scholar
[Sa1]Sacks, G. E., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 8 (1964), pp. 300–312.CrossRefGoogle Scholar
[Sa2]Sacks, G. E., Degrees of unsolvability, Annals of Mathematical Studies, no. 55, Princeton Univ. Press, Princeton, N.J., 1966.Google Scholar