Skip to main content
×
Home
    • Aa
    • Aa

The universal splitting property. II

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

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).

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 1 *
Loading metrics...

Abstract views

Total abstract views: 36 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 30th April 2017. This data will be updated every 24 hours.