## Some orbits for ... (2000)

Venue: | ANN. PURE APPL. LOGIC |

Citations: | 6 - 4 self |

### BibTeX

@ARTICLE{Cholak00someorbits,

author = {Peter Cholak and Rod Downey and Eberhard Herrmann},

title = {Some orbits for ...},

journal = {ANN. PURE APPL. LOGIC},

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

837 |
Theory of recursive functions and effective computability
- Rogers
- 1967
(Show Context)
Citation Context ... pseudo-creative iff for all B disjoint from A, there exists an infinite c.e. set C disjoint from A ∪ B. We remark that if A is a simple set then A × ω is pseudo-creative and r-separable. (See Rogers =-=[26]-=-, Exercise 8.36). As we will see, not every degree contains Herrmann sets. It is not difficult to prove that a set A is Herrmann iff it is D-maximal, r-separable and pseudo-creative. Lemma 2.4 A is st... |

309 | Classical Recursion Theory - Odifreddi - 1989 |

83 | Theory of Recursive Functions and E#ective Computability - Rogers - 1967 |

72 |
Soare, Recursively Enumerable Sets and Degrees
- I
- 1987
(Show Context)
Citation Context ... the two proofs. We remark that in some sense the proofs are kind of the same since there is a hidden use of the extension machinery in the following theorem of Soare. Lemma 5.1 (Soare’s Lemma, Soare =-=[25]-=-) Let Z be c.e. and let S(Z) denote the structure {Y : Y is c.e. and either Y ∪ Z = ω or Y ⊆ ∗ Z} with inclusion relation. Let S ∗ (Z) be S(Z) modulo the finite sets. Then for any two infinite c.e. 17... |

53 | Classes of recursively enumerable sets and degrees of unsolvability - Martin - 1966 |

32 | Automorphisms of the Lattice of Recursively Enumerable Sets
- Cholak
- 1995
(Show Context)
Citation Context ...ich allows us to prove results about automorphisms of E without having to construct effective skeletons 1 (as the original extension lemma needs) nor to apply the whole tree methodology as the Cholak =-=[1]-=- or HarringtonSoare [13] machinery needs. In the next section, we will use Cholak’s version to establish that Herrmann sets form an orbit 2 . First we need some notation and terminology. Definition 3.... |

22 |
Automorphisms of the lattice of recursively enumerable sets, Part I: Maximal sets
- Soare
- 1974
(Show Context)
Citation Context ...numerable set determine its degree. One of the reasons for this lack of understanding is the lack of known orbits for E. So far, the only externally defined orbits we have are the maximal sets (Soare =-=[24]-=-) and variations such as quasimaximal, the hemimaximal sets (Downey-Stob [9]), and the creative sets (Harrington-Myhill see [25]). The goal of the present paper is to extend this collection, extending... |

21 | Post’s Program and incomplete recursively enumerable sets
- Harrington, Soare
- 1991
(Show Context)
Citation Context ...romptness and S(A) relate. We remark that promptness considerations are central to recent investigations about orbits and complete sets (e.g. Cholak, Downey, Stob [4], Cholak [1, 2], Harrington-Soare =-=[12, 13, 14]-=-, Wald [27]). We need the following definition. Definition 8.1 We say a set A is effectively S if A satisfies S(A) and moreover, this is an effective procedure to compute an index for BX from one for ... |

17 |
Splitting theorems in recursion theory
- Downey, Stob
- 1993
(Show Context)
Citation Context ...wo proofs that hemimaximal sets form an orbit. One is due to Downey and Stob [9], and is based on modifying the extension lemma for pairs. The other is due to Herrmann and can be found in Downey-Stob =-=[11]-=-, is much shorter, and relieves on Theorem 5.3. Since it is very short and provides an interesting reflection on Lemma 5.3, we will provide another version here. Theorem 6.3 The hemimaximal sets form ... |

15 |
Recursively enumerable generic sets
- Maass
- 1982
(Show Context)
Citation Context ...ch that if a ′ = c, then there is a set A in a with S(A). 39Proof. (i) It is a routine finite injury argument to construct a promptly simple set N with a semilow complement satisfying S(A). By Maass =-=[19]-=-, such sets are all automorphic, and hence, all promptly simple sets with semilow complements satisfy S(A). Of course all prompt degrees contain promptly simple sets with semilow complements. (ii) Coo... |

14 |
A non-inversion theorem for the jump operator
- Shore
- 1988
(Show Context)
Citation Context ...nd hence, all promptly simple sets with semilow complements satisfy S(A). Of course all prompt degrees contain promptly simple sets with semilow complements. (ii) Cooper [6], and independently, Shore =-=[23]-=- constructed a degree c ̸= 0 ′′ computable enumerable in, and above 0 ′, such that if a ′ = c, then a is prompt. The result follows by (i). □ Another result relating promptness to S(A) is the followin... |

13 | Minimal pairs and high recursively enumerable degrees - Cooper - 1974 |

10 | Codable sets and orbits of computably enumerable sets
- Harrington, Soare
- 1998
(Show Context)
Citation Context ...romptness and S(A) relate. We remark that promptness considerations are central to recent investigations about orbits and complete sets (e.g. Cholak, Downey, Stob [4], Cholak [1, 2], Harrington-Soare =-=[12, 13, 14]-=-, Wald [27]). We need the following definition. Definition 8.1 We say a set A is effectively S if A satisfies S(A) and moreover, this is an effective procedure to compute an index for BX from one for ... |

6 | There is no fat orbit
- Downey, Harrington
- 1996
(Show Context)
Citation Context ... such as jump inversion, highness and the like, and relate their degrees to various known classes such as prompt sets and the hemimaximal sets as well as the degrees analyzed by Downey and Harrington =-=[8]-=- in the “no fat orbit” result. We also obtain additional results on the possible tardiness of Downey-Harrington sets. Our most interesting result in this vein is to show that there is a c.e. degree wh... |

4 |
Automorphisms of the lattice of recursively enumerable sets and hyperhypersimple sets
- Herrmann
- 1989
(Show Context)
Citation Context ...tell, the first author to study L(A)/D(A), and define D-maximal and D-quasimaximal in terms of this structure, even in passing, was Degtev [7], but the first systematic study can be found in Herrmann =-=[15]-=-. We remark that an equivalent formulation of D-quasimaximality is that L(A)/D(A) is finite. Theorem 2.1 L(A)/D(A) is finite iff L(A)/D(A) is a finite boolean algebra. Proof. Suppose that L(A)/D(A) is... |

3 |
A jump class of noncappable degrees
- Cooper
- 1989
(Show Context)
Citation Context ...ch sets are all automorphic, and hence, all promptly simple sets with semilow complements satisfy S(A). Of course all prompt degrees contain promptly simple sets with semilow complements. (ii) Cooper =-=[6]-=-, and independently, Shore [23] constructed a degree c ̸= 0 ′′ computable enumerable in, and above 0 ′, such that if a ′ = c, then a is prompt. The result follows by (i). □ Another result relating pro... |

3 |
Jumps of hemimaximal sets
- Downey, Stob
- 1991
(Show Context)
Citation Context ...e different, they do exhibit a number of similarities. Up to the present paper, the 26hemimaximal degrees were the only known elementary definable orbit realizing all possible jumps (Downey and Stob =-=[10]-=-). Herrmann sets share this property. Theorem 7.3 Let S be c.e. in and above ∅ ′ . Then there is a Herrmann set A with A ′ ≡T S. Proof. The argument is similar to Downey-Stob [10], Theorem 2.1. It is ... |

3 |
Diagonals and Dmaximal sets
- Herrmann, Kummer
- 1994
(Show Context)
Citation Context ...ns of K , (the standard code of the halting set), under some Friedberg enumeration (see Kummer [18]). These sets and their degrees have been previously examined by Kummer [18] and Herrmann and Kummer =-=[17]-=-. Herrmann sets are ones that are D-maximal and have an additional property (strong R-separability). Our proof of the fact that Herrmann sets form an obit, admits some further modifications. For insta... |

2 |
The translation theorem, Archive for
- Cholak
- 1994
(Show Context)
Citation Context ...ng into A are the Ge and Ke. 3 Cholak’s Modified Extension Lemma The beautiful extension theorem of Soare [24] occupies a justifiably central place in the study of the automorphism group of E. Cholak =-=[2]-=- proved a very useful variation on this result which allows us to prove results about automorphisms of E without having to construct effective skeletons 1 (as the original extension lemma needs) nor t... |

2 |
Diagonal and semihyperhypersimple sets
- Kummer
- 1991
(Show Context)
Citation Context ...hyperhypersimple sets are of independent 2interest since they are precisely the sets that are not versions of K , (the standard code of the halting set), under some Friedberg enumeration (see Kummer =-=[18]-=-). These sets and their degrees have been previously examined by Kummer [18] and Herrmann and Kummer [17]. Herrmann sets are ones that are D-maximal and have an additional property (strong R-separabil... |

2 |
Recursion, metarecursion and inclusion
- Owings
- 1967
(Show Context)
Citation Context ...g c.e.) (∀C ⊇ A)(∀X ⊆ A)[(B ∩ C) − A not c.e. ∨ B ∪ C ∪ X ̸= ω], then B can be split into a pair of sets B1 ⊔ B2 = B, both of which have the same property as B. (An analog of Owings splitting theorem =-=[22]-=-.) This is enough for our purposes since we claim that [B1] <D [B] and is also non-complemented, contradicting the minimality of [B]. Since [B1] ≤D [B] it can only be that [B] ≡D [B1]. Suppose that [B... |

1 |
Minimal 1-degrees and truth-table reducibility
- Degtev
- 1976
(Show Context)
Citation Context ...bility proofs of the first order theory of E. As best we can tell, the first author to study L(A)/D(A), and define D-maximal and D-quasimaximal in terms of this structure, even in passing, was Degtev =-=[7]-=-, but the first systematic study can be found in Herrmann [15]. We remark that an equivalent formulation of D-quasimaximality is that L(A)/D(A) is finite. Theorem 2.1 L(A)/D(A) is finite iff L(A)/D(A)... |

1 | The # 3 automorphism method and nonivariant classes of degrees - Harrington, Soare - 1996 |

1 |
Personal communication to the first two authors
- Herrmann
- 1989
(Show Context)
Citation Context ...[9]), and the creative sets (Harrington-Myhill see [25]). The goal of the present paper is to extend this collection, extending and giving proofs of some, more or less, unpublished claims of Herrmann =-=[16]-=-. Our first new orbit is a class of sets we call Herrmann sets, based on the fact that Herrmann was the first to claim that they formed an orbit. Here we give two proofs. The first proof is based on a... |

1 | Automorphism and noninvariant properites of the computably enumerable sets - Wald - 1999 |