## Highness and bounding minimal pairs (1993)

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Venue: | Math. Logic Quarterly |

Citations: | 3 - 2 self |

### BibTeX

@ARTICLE{Downey93highnessand,

author = {Rodney G. Downey and Richard A. Shore},

title = {Highness and bounding minimal pairs},

journal = {Math. Logic Quarterly},

year = {1993},

pages = {475--491}

}

### OpenURL

### Abstract

We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high2 nonbounding r.e. degree. 0

### Citations

53 |
Classes of recursively enumerable sets and degrees of unsolvability
- Martin
- 1966
(Show Context)
Citation Context ...acks[16] Splitting Theorem in which d is recursive.) For the high sets (A ′ ≡T ∅ ′′ ) and degrees, the trend of results has been that anything possible happens. Here the classic examples are Martin’s =-=[14]-=- theorem that every high degree contains a maximal set and Cooper’s [2] result that there is a minimal pair (a,b ̸= 0 with a ∧ b = 0) below every high degree. More recently, Shore and Slaman [19] and ... |

45 |
A minimal degree less than 0
- Sacks
- 1961
(Show Context)
Citation Context ...g like the recursive one. A classic example is the Robinson [15] Splitting Theorem: If d < c and d is low then there are r.e. a and b with d < a, b < c such that a ∨ b = c. (This generalizes the Sacks=-=[16]-=- Splitting Theorem in which d is recursive.) For the high sets (A ′ ≡T ∅ ′′ ) and degrees, the trend of results has been that anything possible happens. Here the classic examples are Martin’s [14] the... |

32 | Automorphisms of the Lattice of Recursively Enumerable Sets
- Cholak
- 1995
(Show Context)
Citation Context ...0] have shown that other important phenomena (the special triples of Slaman[21] and the nonsplitting pairs of Lachlan [12], respectively) occur below every high degree. As for the lattice E ∗ , Cholak=-=[1]-=- and Harrington and Soare [10] have proven that every possible lattice of supersets L ∗ (A) occurs as one of a high set B, i.e. there is a high B with L ∗ (A) ∼ = L ∗ (B). Indeed, if A is not recursiv... |

27 |
Interpolation and embedding in the recursively enumerable degrees
- ROBINSON
- 1971
(Show Context)
Citation Context .... ‡ Partially supported by NSF Grants DMS-9204308 and INT 90-20558and ARO through MSI, Cornell University, DAAL-03-C-0027. 1degrees behaving like the recursive one. A classic example is the Robinson =-=[15]-=- Splitting Theorem: If d < c and d is low then there are r.e. a and b with d < a, b < c such that a ∨ b = c. (This generalizes the Sacks[16] Splitting Theorem in which d is recursive.) For the high se... |

26 |
A recursively enumerable degree which will not split over all lesser ones
- Lachlan
- 1975
(Show Context)
Citation Context ...th a ∧ b = 0) below every high degree. More recently, Shore and Slaman [19] and [20] have shown that other important phenomena (the special triples of Slaman[21] and the nonsplitting pairs of Lachlan =-=[12]-=-, respectively) occur below every high degree. As for the lattice E ∗ , Cholak[1] and Harrington and Soare [10] have proven that every possible lattice of supersets L ∗ (A) occurs as one of a high set... |

14 |
Degrees of recursively enumerable sets which have no maximal supersets
- Lachlan
(Show Context)
Citation Context ...high and low2 degrees. The high ones, for example, are precisely the ones containing maximal sets (Martin [14]). The low2 degrees are precisely those containing sets with no maximal superset (Lachlan =-=[11]-=- and Shoenfield [17]). In R, the above mentioned results of Shore and Slaman [18], [19] have separated these two classes. More recently,Groszek and Slaman [8] have combined Lachlan’s nonbounding theor... |

14 |
Working below a high recursively enumerable degree
- Shore, Slaman
(Show Context)
Citation Context ...in’s [14] theorem that every high degree contains a maximal set and Cooper’s [2] result that there is a minimal pair (a,b ̸= 0 with a ∧ b = 0) below every high degree. More recently, Shore and Slaman =-=[19]-=- and [20] have shown that other important phenomena (the special triples of Slaman[21] and the nonsplitting pairs of Lachlan [12], respectively) occur below every high degree. As for the lattice E ∗ ,... |

13 |
Minimal pairs and high recursively enumerable degrees
- Cooper
- 1974
(Show Context)
Citation Context ...A ′ ≡T ∅ ′′ ) and degrees, the trend of results has been that anything possible happens. Here the classic examples are Martin’s [14] theorem that every high degree contains a maximal set and Cooper’s =-=[2]-=- result that there is a minimal pair (a,b ̸= 0 with a ∧ b = 0) below every high degree. More recently, Shore and Slaman [19] and [20] have shown that other important phenomena (the special triples of ... |

10 |
Working below a low2 recursively enumerable degree, Archive for Mathematical Logic 29
- Shore, Slaman
- 1990
(Show Context)
Citation Context ...to B. There has been some recent work extending such results on low sets to low2 ones (A ′′ ≡T ∅ ′′ ). Harrington et al. [9] have shown that if A is low2 then L ∗ (A) ∼ = E ∗ . In R, Shore and Slaman =-=[18]-=- have shown that all extensions of embedding not ruled out by two classical theorems can be done in the low2 r.e. degrees. (They also supply a proof of Harrington’s extension of the Robinson splitting... |

9 |
Bounding minimal pairs
- Lachlan
- 1979
(Show Context)
Citation Context ...d Shoenfield [17]). In R, the above mentioned results of Shore and Slaman [18], [19] have separated these two classes. More recently,Groszek and Slaman [8] have combined Lachlan’s nonbounding theorem =-=[13]-=- (there is a nonzero degree with no minimal pair below it) with Lachlan’s [12] nonsplitting theorem to provide a definable class that is disjoint from both the high and low2 degrees. There are, howeve... |

5 |
1990a], Notes on the 0 priority method with special attention to density results
- Downey
- 1990
(Show Context)
Citation Context ...o guess as to whether they are truly visited or not. Thus the above technique for making a set high2 cannot be combined with the link mechanism of the construction [7]. The reader is referred to [7], =-=[3]-=-, and [4] for further details. 7 Other High2 Applications. The technique of Section 5 can be applied in other situations. To illustrate this, we give one further example. We shall answer a question fr... |

4 |
Degrees of classes of r.e
- Shoenfield
- 1976
(Show Context)
Citation Context ...s. The high ones, for example, are precisely the ones containing maximal sets (Martin [14]). The low2 degrees are precisely those containing sets with no maximal superset (Lachlan [11] and Shoenfield =-=[17]-=-). In R, the above mentioned results of Shore and Slaman [18], [19] have separated these two classes. More recently,Groszek and Slaman [8] have combined Lachlan’s nonbounding theorem [13] (there is a ... |

3 |
Jumps of hemimaximal sets
- Downey, Stob
- 1991
(Show Context)
Citation Context ...mimaximal sets (halves of splittings of maximal sets). (On the other hand, Downey and Stob [5] show that every high degree contains such a set.) This application answers a question of Downey and Stob =-=[6]-=- and [6]. On the low side of the jump hierarchy, Shore and Slaman [19] show that the Slaman triples can have a low3 top. Taken together, these results indicate that the classes defined by Slaman and G... |

3 | Soare, "Recursively enumerable sets and degrees - I - 1987 |

3 |
1983] The recursively enumerable degrees as a substructure of the degrees, handwritten notes
- Slaman
(Show Context)
Citation Context ...t that there is a minimal pair (a,b ̸= 0 with a ∧ b = 0) below every high degree. More recently, Shore and Slaman [19] and [20] have shown that other important phenomena (the special triples of Slaman=-=[21]-=- and the nonsplitting pairs of Lachlan [12], respectively) occur below every high degree. As for the lattice E ∗ , Cholak[1] and Harrington and Soare [10] have proven that every possible lattice of su... |

1 |
Density and infima in the recursively enumerable degrees, in preparation
- Downey, Shore
(Show Context)
Citation Context ...provide a definable class that is disjoint from both the high and low2 degrees. There are, however, no definitions or characterizations of any of the jump classes in R. (In contrast, Downey and Shore =-=[4]-=- have actually defined the low2 r.e. sets in Rtt, the structure of the r.e. truth table degrees, as precisely those with minimal covers in Rtt.) The obvious general problem is to give order theoretic ... |

1 |
Minimal pairs
- Downey, Stob
(Show Context)
Citation Context ...eliminate one of the requirements of globally higher priority. In this way, we will be certain that we get a loss on only finitely many m’s. 18We will employ the following technique from Downey-Stob =-=[7]-=-. When we hit τ, we realize that if there is a link from τ down then this may be a potentially permanent link. To make the combinatorics easier, instead of directly going to σ, we first do a scouting ... |

1 |
Soare, New methods for automorphisms of the recursively enumerable sets and applications to the low 2 sets, in preparation
- Harrington, Lachlan, et al.
(Show Context)
Citation Context ...ed, if A is not recursive, they construct an automorphism of E ∗ which takes A to B. There has been some recent work extending such results on low sets to low2 ones (A ′′ ≡T ∅ ′′ ). Harrington et al. =-=[9]-=- have shown that if A is low2 then L ∗ (A) ∼ = E ∗ . In R, Shore and Slaman [18] have shown that all extensions of embedding not ruled out by two classical theorems can be done in the low2 r.e. degree... |

1 | Soare, A solution of Post's program; r.e. sets automorphic to complete sets; and new noninvariant jump classes of r,e, degrees, in preparation - Harrington, I |

1 |
Splitting and density cannot be combined below a high recursively enumerable degree, in preparation
- Shore, Slaman
(Show Context)
Citation Context ... theorem that every high degree contains a maximal set and Cooper’s [2] result that there is a minimal pair (a,b ̸= 0 with a ∧ b = 0) below every high degree. More recently, Shore and Slaman [19] and =-=[20]-=- have shown that other important phenomena (the special triples of Slaman[21] and the nonsplitting pairs of Lachlan [12], respectively) occur below every high degree. As for the lattice E ∗ , Cholak[1... |

1 | The recursively enumerable degrees as a substructure of the \Delta 2 degrees - Slaman |

1 |
Tree arguments in recursion theory and the 0 ′′′ priority method
- Soare
- 1985
(Show Context)
Citation Context ...structions we shall look only at properties can be possessed by high2 but cannot be possessed by high degrees. In this section, we review the nonbounding theorem of Lachlan [13] as presented in Soare =-=[22]-=- and [23]. Theorem 4.1 . There is a high2 recursively enumerable degree c that bounds no minimal pair. Proof. Since the construction of a nonbounding degree is a very well analyzed and documented resu... |