## On the Universal Splitting Property (1996)

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Venue: | Mathematical Logic Quarterly |

Citations: | 3 - 2 self |

### BibTeX

@INPROCEEDINGS{Downey96onthe,

author = {Rod Downey},

title = {On the Universal Splitting Property},

booktitle = {Mathematical Logic Quarterly},

year = {1996},

pages = {43}

}

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### Abstract

We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from Ambos-Spies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 are said to split A if A = A 1 [ A 2 and A 1 " A 2 = ;. We write A 1 t A 2 = A in the case that A 1 and A 2 split A. Splitting theorems for computably enumerable sets have played a central role in the history of classical computability theory. For instance, Sack's splitting theorm [14], demonstrated that every nonzero computably enumerable degree could be Downey's research supported by Cornell University, an IGC grant from Victoria University and the New Zealand Marsden Fund via grant 95-VIC-MIS-0698 under contract VIC-509. Some of these results were obtained whilst Downey was a Visiting Professor at Cornell University in fall 1995. decomposed into a pair of incomparible nonzero computa...

### Citations

324 | Classical recursion theory - Odifreddi - 1999 |

80 |
Soare, Recursively enumerable sets and degrees
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(Show Context)
Citation Context ...his stage we do not know if Theorem 1.2 can be improved to say that there is a non wtt-topped degree containing a set with the universal splitting property. Our notation is standard and follows Soare =-=[15]-=-. We remind the reader that all uses etc at stage s are bounded by s and are nondecreasing in both argument and stage number. The hat convention applies throughout. 3 2 Proof of Theorem 1.1 Recall tha... |

48 |
On the degrees less than 0
- Sacks
- 1963
(Show Context)
Citation Context ...the case that A 1 and A 2 split A. Splitting theorems for computably enumerable sets have played a central role in the history of classical computability theory. For instance, Sack's splitting theorm =-=[14]-=-, demonstrated that every nonzero computably enumerable degree could be Downey's research supported by Cornell University, an IGC grant from Victoria University and the New Zealand Marsden Fund via gr... |

32 | Automorphisms of the Lattice of Recursively Enumerable Sets - Cholak - 1995 |

19 |
The weak truth table degrees of recursively enumerable sets
- Ladner, Sasso
- 1975
(Show Context)
Citation Context ... the degrees with such sets. 2 The original construction in Lerman and Remmel [12] of a set with the universal splitting property resembled the construction of a contiguous degree in Ladner and Sasso =-=[10]-=-. (Recall that a degree is contiguous Turing degree is one containing a single computably enumerable wtt-degree.) The connection between the wtt-degree structure of a T-degree was futher noted by Ambo... |

17 |
Splitting theorems in recursion theory
- Downey, Stob
- 1993
(Show Context)
Citation Context ...Abstract We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from Ambos-Spies and Fejer [1] and Downey and Stob =-=[7]. Som-=-e technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 are said to split A if A = A 1 [ A 2 and A 1 " A 2 = ;. We write A 1 t A 2 = A in the case that A... |

12 |
jump classes and strong reducibilities
- Downey, Jockusch
- 1987
(Show Context)
Citation Context ...e was futher noted by Ambos-Spies and Fejer who observed that if is A has the property that for all BsT A, B wtt A then A \Theta ! has the universal splitting property. (Following Downey and Jockusch =-=[5]-=-, we call deg(A) wtt-topped with top A.) It is known that all incomplete wtttopped degrees are low 2 , and all contiguous degrees are low 2 . Recently Downey and Lempp [6] demonstrated that the contig... |

6 | Postsâ€™s program and incomplete recursively enumerable sets - Harrington, Soare - 1991 |

6 |
The universal splitting property
- Lerman, Remmel
- 1984
(Show Context)
Citation Context ...y enumerable degrees. S(A) = fdeg(A 1 ) : (9A 2 )[A 1 t A 2 = A]g; and, S 2 (A) = fhdeg(A 1 ); deg(A 2 )i : A 1 t A 2 = Ag: In this paper, our concerns are two questions implicit in Lerman and Remmel =-=[11, 12]-=- and Ambos-Spies and Fejer [1], about the the property known as the universal splitting property. By Sacks splitting theorem, we know that S(A) and S 2 (A) have infinitely many elements. The largest p... |

4 |
The degrees of r.e. sets without the universal splitting property
- Downey
- 1985
(Show Context)
Citation Context ... A has the strong universal splitting property if S 2 (A) is as large as possible. There have been quite a number of results concerning the (strong) universal splitting property. For instance, Downey =-=[3]-=- proved that every nonzero computably enumerable degree degree contained a computably enumerable set without ithe universal splitting property, and indeed, in [4], proved that no hypersimple computabl... |

3 |
Subsets of hypersimple sets
- Downey
- 1987
(Show Context)
Citation Context ...itting property. For instance, Downey [3] proved that every nonzero computably enumerable degree degree contained a computably enumerable set without ithe universal splitting property, and indeed, in =-=[4]-=-, proved that no hypersimple computably enumerable has the universal splitting property. (This is a kind of splitting analogue to Stob's result [16] that a computably enumerable set is simple iff it d... |

3 |
Jockusch Jr., Relationships between reducibilities
- G
- 1969
(Show Context)
Citation Context ...sal splitting property then L = fe : W esT Ag will be a \Sigma 0 3 set. (e 2 L iff W f(e) wtt A. And fj : W j wtt Ag is \Sigma 0 3 .) But then A is low 2 since fe : W esT Ag is \Sigma A 3 . (Jockusch =-=[9]-=-.) Proof of Lemma 2.1. Let B be a given dump set. Suppose that BsT A via a reduction \Phi A = B, yet B 6 wtt A. We construct a set CsT A via a reduction \Delta A = C to meet the requirements below. R ... |

2 |
theoretic splitting properties of recursively enumerable sets
- Ambos-Spies, Fejer
- 1988
(Show Context)
Citation Context ...ealand February 9, 1996 Abstract We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from Ambos-Spies and Fejer =-=[1] and -=-Downey and Stob [7]. Some technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 are said to split A if A = A 1 [ A 2 and A 1 " A 2 = ;. We write A 1 t A ... |

1 |
Contiguity and distributivity in the enumerable degrees, submitted
- Downey, Lempp
(Show Context)
Citation Context ...ollowing Downey and Jockusch [5], we call deg(A) wtt-topped with top A.) It is known that all incomplete wtttopped degrees are low 2 , and all contiguous degrees are low 2 . Recently Downey and Lempp =-=[6]-=- demonstrated that the contiguous degrees are definable in R. By the work of Ambos-Spies and Fejer, [1], Proposition 2.1, the Downey-Lemmp definition also proved that a computably enumerable degree is... |

1 |
Index sets and degrees of unsolvability
- Stob
- 1982
(Show Context)
Citation Context ...ut ithe universal splitting property, and indeed, in [4], proved that no hypersimple computably enumerable has the universal splitting property. (This is a kind of splitting analogue to Stob's result =-=[16]-=- that a computably enumerable set is simple iff it does not have computably enumerable supersets of each nonzero degree. The Downey hypersimple set result says that the position of a computable enumer... |

1 | Ambos-Spies and P.A.Fejer, Degree theoretical splitting properties of recursively enumerable sets - unknown authors |

1 | Automorphisms of the LatticeofRecursively Enumerable Sets - Cholak - 1995 |

1 | The universal splitting propertyI,inD.van - Lerman, Remmel - 1982 |