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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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Cited by 17 (6 self)
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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.
There is no Fat Orbit
 Ann. Pure Appl. Logic
"... We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions. ..."
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Cited by 6 (3 self)
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We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.
On the Universal Splitting Property
 Mathematical Logic Quarterly
, 1996
"... We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from AmbosSpies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 ar ..."
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Cited by 2 (2 self)
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We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from AmbosSpies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 are said to split A if A = A 1 [ A 2 and A 1 " A 2 = ;. We write A 1 t A 2 = A in the case that A 1 and A 2 split A. Splitting theorems for computably enumerable sets have played a central role in the history of classical computability theory. For instance, Sack's splitting theorm [14], demonstrated that every nonzero computably enumerable degree could be Downey's research supported by Cornell University, an IGC grant from Victoria University and the New Zealand Marsden Fund via grant 95VICMIS0698 under contract VIC509. Some of these results were obtained whilst Downey was a Visiting Professor at Cornell University in fall 1995. decomposed into a pair of incomparible nonzero computa...
Simple Sets Are Not BttCuppable
, 1997
"... We extend Post's result that a simple set cannot be bttcomplete by showing that in fact it cannot be bttcuppable, i.e. if the join of a c.e. set and a simple set is bttcomplete, then the nonsimple set is bttcomplete itself. The proof also yields that simple sets are not dcuppable (i.e. no ..."
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Cited by 1 (1 self)
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We extend Post's result that a simple set cannot be bttcomplete by showing that in fact it cannot be bttcuppable, i.e. if the join of a c.e. set and a simple set is bttcomplete, then the nonsimple set is bttcomplete itself. The proof also yields that simple sets are not dcuppable (i.e. not cuppable with regard to disjunctive reductions). Post showed that a simple set cannot be bttcomplete. In a paper by Stephen Fenner and the author [3] this was generalized to nonc.e. sets by isolating the immunity property which is responsible for the incompleteness. Another approach to the bttincompleteness of simple sets would have been through degrees. How incomplete are simple sets? Putting it dioeerently: can the join of a bttincomplete degree with a simple degree be bttcomplete? We will show that the answer is no. Deønition 1 A set A is called rcuppable, if there is a c.e. set B such that ; 0 r A \Phi B and ; 0 6 r B, where r is a class of reductions (like m, 1, btt, c, d, tt...
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 Pi Classes
, 2006
"... This paper continues the study of the lattice of \Pi ..."