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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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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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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.
THE UPWARD CLOSURE OF A PERFECT THIN CLASS
"... Abstract. There is a perfect thin Π01 class whose upward closure in the Turing degrees has full measure (and indeed contains every 2random degree.) Thus, in the Muchnik lattice of Π01 classes, the degree of 2random reals is comparable with the degree of some perfect thin class. This solves a quest ..."
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Abstract. There is a perfect thin Π01 class whose upward closure in the Turing degrees has full measure (and indeed contains every 2random degree.) Thus, in the Muchnik lattice of Π01 classes, the degree of 2random reals is comparable with the degree of some perfect thin class. This solves a question of Simpson [16]. 1.
Invariance And Noninvariance In The Lattice Of ... Classes
"... We prove that there are two minimal 1 classes that are not automorphic. ..."
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.
Invariance And Noninvariance In The Lattice Of Pi Classes
, 2006
"... This paper continues the study of the lattice of \Pi ..."
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, 2007
"... This thesis analyzes the structure of the Medvedev lattice of nonempty Π 0 1 classes in 2 ω from the viewpoint of branching and nonbranching degrees. This lattice is a countable distributive lattice with least and greatest element, which describes the relative information content of certain subset ..."
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This thesis analyzes the structure of the Medvedev lattice of nonempty Π 0 1 classes in 2 ω from the viewpoint of branching and nonbranching degrees. This lattice is a countable distributive lattice with least and greatest element, which describes the relative information content of certain subsets of 2 ω. Chapter 1 is an introduction, providing background history, notation, and an overview of necessary concepts. Chapter 2 is essentially my paper “NonBranching Degrees in the Medvedev Lattice of Π 0 1 classes.”[1]. The chapter adds an additional theorem which strengthens the theorem on inseparable and not hyperinseparable classes. The chapter is also slightly more verbose. We begin by taking an existing condition, homogeneous, which implies nonbranching and define two successively weaker conditions, hyperinseparable and inseparable. We then demonstrate that inseparable is equivalent to nonbranching and is invariant under Medvedev equivalence. Finally, we prove separation theorems, namely the existence of an inseparable and not hyperinseparable degree and the existence of a hyperinseparable and not homogeneous degree. Chapter 3 defines a combinatorial method for constructing Π 0 1 classes by priority arguments. This section does not contain any difficult proofs but abstracts many of the common elements of such constructions. The definitions and results are used in