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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 ... Classes; Perfect Thin Classes and Anc Degrees
, 1999
"... \Pi 0 1 classes are important to the logical analysis of many parts of mathematics. The \Pi 0 1 classes form a lattice. As with the lattice of computable enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximalit ..."
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\Pi 0 1 classes are important to the logical analysis of many parts of mathematics. The \Pi 0 1 classes form a lattice. As with the lattice of computable enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality namely the notion of a thin class. We prove a number of results relating automorphisms, invariance and thin classes. Our main result is an analog of the MartinSoare work on maximal sets and high degrees, using thin classes and anc degrees. In particular, we show that the perfect thin classes are definable (in the lattice of \Pi 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 \Pi 0 1 classes. We show that all perfect thin classes are automorphic (via a \Delta 0 3 automorphism). 1 Introduction In [19] Post was the first to articulate the connection between properties of the lattice...
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 ..."
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.