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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.
Countable thin Π0 1 classes
 Annals of Pure and Applied Logic
, 1993
"... Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive CantorBendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is ..."
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Abstract. AΠ0 1 class P ⊂ {0,1}ωis thin if every Π0 1 subclass Q of P is the intersection of P with some clopen set. Countable thin Π0 1 classes are constructed having arbitrary recursive CantorBendixson rank. A thin Π0 1 class P is constructed with a unique nonisolated point A of degree 0 ′. It is shown that, for all ordinals α>1, no set of degree ≥ 0 ′ ′ can be a member of any thin Π0 1 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π0 1 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree (not necessarily r.e.) that contains a set which is of rank one in some thin Π0 1 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π0 1 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π0 1 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree 0 ′ ′. Introduction.
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.
IDEALS IN COMPUTABLE RINGS
"... Abstract. We show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a field is equivalent to WKL0 over RCA0, and that the existence of a nontrivial proper finitely generated ideal in a commutative ring with identity which is not a field is equivalent to ..."
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Abstract. We show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a field is equivalent to WKL0 over RCA0, and that the existence of a nontrivial proper finitely generated ideal in a commutative ring with identity which is not a field is equivalent to ACA0 over RCA0. We also prove that there are computable commutative rings with identity where the nilradical is Σ 0 1complete, and the Jacobson radical is Π 0 2complete, respectively. 1.
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...
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.
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 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. ..."
Index sets for . . .
, 1997
"... ... class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of ..."
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... class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of