The Journal of Symbolic Logic

Research Article

The universal splitting property. II

M. Lermana11 and J. B. Remmela22

a1 University of Connecticut, Storrs, Connecticut 06068

a2 University of California at San Diego, La Jolla, California 92093

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 B T A and C T A). 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 < T A and C < T A. 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 T D).

(Received February 03 1982)

(Revised April 03 1982)

Footnotes

1   Research supported by the National Science Foundation under grant number MCS 78-01849

2   Research supported by the National Science Foundation under grant number MCS 79-03406