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Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees
 Trans. Amer. Math. Soc
"... Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 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, o ..."
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Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 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 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 collection of perfect thin classes (a notion which is definable in the lattice of Π0 1 classes) forms an orbit in the lattice of Π01 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 Π0 1 classes. We remark that the automorphism result is proven via a ∆0 3 automorphism, and demonstrate that this complexity is necessary. 1.
Slender classes
, 2006
"... A Π 0 1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩ C. Cholak, Coles, Downey and Herrmann [7] proved that a Π 0 1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is ..."
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Cited by 2 (1 self)
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A Π 0 1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩ C. Cholak, Coles, Downey and Herrmann [7] proved that a Π 0 1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is the free Boolean algebra, then all such think classes are automorphic in the lattice of Π 0 1 classes under inclusion. From this it follows that if the boolean algebra has a finite number n of atoms then the resulting classes are all automorphic. We prove a conjecture of Cholak and Downey [8] by showing that this is the only time the Boolean algebra determines the automorphism type of a thin class.
AUTOMORPHISMS OF THE LATTICE OF �0 1 CLASSES; PERFECT THIN CLASSES AND ANC DEGREES
"... 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, o ..."
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Cited by 1 (1 self)
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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.
Jump inversions inside effectively closed sets and applications to randomness
 J. Symbolic Logic
"... Abstract. We study inversions of the jump operator on Π0 1 classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2random ..."
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Cited by 1 (1 self)
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Abstract. We study inversions of the jump operator on Π0 1 classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2random sets which are not 2random, and the jumps of the weakly 1random relative to 0 ′ sets which are not 2random. Both of the classes coincide with the degrees above 0 ′ which are not 0 ′dominated. A further application is the complete solution of [Nie09, Problem 3.6.9]: one direction of van Lambalgen’s theorem holds for weak 2randomness, while the other fails. Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [Nie09, Problem 8.2.14]: not all weakly 2random sets are array computable. In fact, given any oracle X, there is a weakly 2random which is not array computable relative to X. This contrasts with the fact that all 2random sets are array computable. 1.
INVARIANCE AND NONINVARIANCE IN THE LATTICE OF � 0 1 CLASSES
"... Abstract. We prove that there are two minimal �0 1 classes that are not automorphic. ..."
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Abstract. We prove that there are two minimal �0 1 classes that are not automorphic.
THE UPWARD CLOSURE OF A PERFECT THIN CLASS
"... Abstract. There is a perfect thin Π0 1 class whose upward closure in the Turing degrees has full measure (and indeed contains every 2random degree.) Thus, in the Muchnik lattice of Π0 1 classes, the degree of 2random reals is comparable with the degree of some perfect thin class. This solves a que ..."
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Abstract. There is a perfect thin Π0 1 class whose upward closure in the Turing degrees has full measure (and indeed contains every 2random degree.) Thus, in the Muchnik lattice of Π0 1 classes, the degree of 2random reals is comparable with the degree of some perfect thin class. This solves a question of Simpson [15]. 1.
Invariance And Noninvariance In The Lattice Of Pi Classes
, 2006
"... This paper continues the study of the lattice of \Pi ..."