## AUTOMORPHISMS OF THE LATTICE OF �0 1 CLASSES; PERFECT THIN CLASSES AND ANC DEGREES

Citations: | 1 - 1 self |

### BibTeX

@MISC{Cholak_automorphismsof,

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

title = {AUTOMORPHISMS OF THE LATTICE OF �0 1 CLASSES; PERFECT THIN CLASSES AND ANC DEGREES},

year = {}

}

### OpenURL

### Abstract

Abstract. �0 1 classes are important to the logical analysis of many parts of mathematics. The �0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance and thin classes. Our main results are an analog of the Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the the collection of perfect thin classes (a notion which is definable in the lattice of �0 1 classes) form an orbit in the lattice of �0 1 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of �0 1 classes. We remark that the automorphism result is proven via a �0 3 automorphism, and demonstrate that this complexity is necessary. 1.

### Citations

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- Cholak
- 1995
(Show Context)
Citation Context ...27]) conjecture holds: Open Question. Is {〈e, i〉 : Pe automorphic to Pi} � 1 1 complete? The analogous result for the lattice of computably enumerable sets was proven by Cholak, Downey and Harrington =-=[6]-=-. 5. Perfect Thin Classes We now turn to perfect classes, our main concern. Recall that for a topological space X, the set X d of derived points of X is the set of x such that x ∈ cl(X − {x}). If x ∈ ... |

19 |
Π 0 1 classes in mathematics
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(Show Context)
Citation Context ...mportant connections more explicit in the next section.) For recent extensive surveys on �0 1 classes and their applications, we refer the reader to Cenzer [2], Cenzer-Jockusch [4], and Cenzer-Remmel =-=[5]-=-. The collection of �0 1 classes form a lattice L(2ω ). In this paper we study this lattice and its connection with the computably enumerable degrees, along the same lines as the well known Post progr... |

17 | On presentations of algebraic structures - Downey - 1997 |

11 | Abstract Dependence, Recursion Theory and the Lattice of Recursively Enumerable Filters - Downey - 1982 |

9 |
Π 0 1-Classes-Structure and applications, in Computability Theory and its
- Cenzer, Jockusch
(Show Context)
Citation Context ...ll make some of these important connections more explicit in the next section.) For recent extensive surveys on �0 1 classes and their applications, we refer the reader to Cenzer [2], Cenzer-Jockusch =-=[4]-=-, and Cenzer-Remmel [5]. The collection of �0 1 classes form a lattice L(2ω ). In this paper we study this lattice and its connection with the computably enumerable degrees, along the same lines as th... |

7 |
Π 0 1 classes in computability theory
- Cenzer
- 1999
(Show Context)
Citation Context ...NN all standard we will make some of these important connections more explicit in the next section.) For recent extensive surveys on �0 1 classes and their applications, we refer the reader to Cenzer =-=[2]-=-, Cenzer-Jockusch [4], and Cenzer-Remmel [5]. The collection of �0 1 classes form a lattice L(2ω ). In this paper we study this lattice and its connection with the computably enumerable degrees, along... |

6 | Effective presentability of Boolean algebras of Cantor–Bendixson rank 1
- Downey, Jockusch
- 1999
(Show Context)
Citation Context ...bove). This would require significant technology since there are computable Boolean algebras that are not even arithmetically isomorphic. This is even true of rank 1 boolean algebras (Downey-Jockusch =-=[11]-=-). A good test case is to consider whether when B1 and B2 are computable copies of the Boolean algebra of finite and cofinite sets, are B1 and B2 automorphic 3 . The proof technique from Theorem 7.6 i... |

3 |
Array nonrecursice sets and multiple permitting arguments
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(Show Context)
Citation Context ... is each follower needs more permissions than its predecessor for followers of the same requirement. Eventually a new degree class, called the anc degrees, was introduced by Downey, Jockusch and Stob =-=[12]-=- to explain such arguments. It turns out that the anc degrees are a class containing all nonlow2 degrees and are closed upwards. They are exactly the degrees realized by many known constructions. We r... |

2 |
Countable thin �0 1 classes
- Cenzer, Downey, et al.
- 1993
(Show Context)
Citation Context ...e. � We remark that it is Open Question 6.3 in Cenzer-Jockusch [4] if “T is finite” is definable in L(2ω ). For more results and background on thin �0 1-classes see Cenzer, Downey, Jockusch and Shore =-=[3]-=- and Downey [9]. The following says that, in a sense, thin classes are the precise analogues of hyperhyper-simple c.e. sets. Lemma 4.3. A nonempty � 0 1 -class P is thin if and only if L(2ω )(P, ↓) is... |

2 |
Maximal theories, Annals of Pure and Applied Logic 33
- Downey
- 1987
(Show Context)
Citation Context ...re we say C is thin if it is infinite and for all � 0 1 subclasses C′ there is a clopen U such that C ′ = C ∩U. What are the basic degree theoretical properties of thin classes. In his thesis, Downey =-=[9]-=- proved that not every degree contains a Martin–Pour-El theory. He showed that while all high degrees contained Martin–Pour-El theories, and some low degrees, there were initial segments not containin... |