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The universal splitting property. II

  • M. Lerman (a1) and J. B. Remmel (a2)

We say that a pair of r.e. sets B and C split an r.e. set A if BC = ∅ and BC = 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 BTA and CTA). 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 DT A, there is a splitting B and C of A such that B and D are Turing equivalent (written BTD).

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[Lac] A. H. Lachlan , The priority method. I, Zeitschrift fur Mathematische Logik und Grundlager d, Mathematik, vol. 13 (1967), pp. 110.

[Lad2] R. E. Ladner , A completely mitotic nonrecursive r.e. degree, Transactions of the America Mathematical Society, vol. 184 (1973), pp. 479507.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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