Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-17T19:45:52.799Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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