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Automorphisms of the lattice of recursively enumerable sets: Orbits, Adv
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to
Degree theoretic definitions of the low_2 recursively enumerable sets
 J. SYMBOLIC LOGIC
, 1995
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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.
The complexity of orbits of computably enumerable sets
 BULLETIN OF SYMBOLIC LOGIC
, 2008
"... The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; ..."
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The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of E; for all finite α ≥ 9, there is a properly ∆0 α orbit (from the proof).
Atomless rmaximal sets
 Israel J. Math
"... Abstract. We focus on L(A), the filter of supersets of A in the structure of the computably enumerable sets under the inclusion relation, where A is an atomless rmaximal set. We answer a long standing question by showing that there are infinitely many pairwise nonisomorphic filters of this type. 1 ..."
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Abstract. We focus on L(A), the filter of supersets of A in the structure of the computably enumerable sets under the inclusion relation, where A is an atomless rmaximal set. We answer a long standing question by showing that there are infinitely many pairwise nonisomorphic filters of this type. 1.
Sorbi A., Algebraic properties of Rogers semilattices of arithmetical numberings
 In Computability and Models, S.B. Cooper and S.S. Goncharov eds.—Kluwer
, 2003
"... Abstract We investigate initial segments and intervals of Rogers semilattices of arithmetical families. We prove that there exist intervals with different algebraic properties; the elementary theory of any Rogers semilattice at arithmetical level n ≥ 2 is hereditarily undecidable; the class of all R ..."
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Abstract We investigate initial segments and intervals of Rogers semilattices of arithmetical families. We prove that there exist intervals with different algebraic properties; the elementary theory of any Rogers semilattice at arithmetical level n ≥ 2 is hereditarily undecidable; the class of all Rogers semilattices of a fixed level n ≥ 2 has an incomplete theory. ∗All authors were partially supported by grant INTASRFBR Computability and Models no.
Immunity Properties and the nC.E. Hierarchy
 in Theory and Applications of Models of Computation, Third International Conference on Computation and Logic, TAMC 2006, Beijing, May 2006, Proceedings, (JinYi
"... Abstract. We extend Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets, and characterise, in the spirit of Post [9], the degrees of the immune and hyperimmune d.c.e. sets. We also show that no properly d.c.e. set can be hhimmune, and indicate how to generalise these results to nc ..."
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Abstract. We extend Post’s programme to finite levels of the Ershov hierarchy of ∆2 sets, and characterise, in the spirit of Post [9], the degrees of the immune and hyperimmune d.c.e. sets. We also show that no properly d.c.e. set can be hhimmune, and indicate how to generalise these results to nc.e. sets, n> 2. 1
On Lachlan's major subdegree problem
 IN: SET THEORY AND THE CONTINUUM, PROCEEDINGS OF WORKSHOP ON SET THEORY AND THE CONTINUUM
, 1989
"... The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become a longstanding open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the ..."
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The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become a longstanding open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b> a if for any c.e. degree x, 0 ′ = b ∨ x if and only if 0 ′ = a ∨ x. In this paper, we show that every c.e. degree b ̸ = 0 or 0 ′ has a major subdegree, answering Lachlan’s question affirmatively. 1
On The Filter Of Computably Enumerable Supersets Of An RMaximal Set
 Arch. Math. Logic
, 2001
"... . We study the filter L (A) of computably enumerable supersets (modulo finite sets) of an rmaximal set A and show that, for some such set A, the property of being cofinite in L (A) is still \Sigma 0 3 complete. This implies that for this A, there is no uniformly computably enumerable " ..."
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. We study the filter L (A) of computably enumerable supersets (modulo finite sets) of an rmaximal set A and show that, for some such set A, the property of being cofinite in L (A) is still \Sigma 0 3 complete. This implies that for this A, there is no uniformly computably enumerable "tower" of sets exhausting exactly the coinfinite sets in L (A). 1. The theorem The computably enumerable (or recursively enumerable) sets form a countable sublattice (denoted by E in the following) of the power set P(!) of the set of natural numbers. The operations of union and intersection are effective on E (i.e., effective in the indices of the computably enumerable sets). The complemented elements of E are exactly the computable sets. The finite sets in E are definable as those elements only bounding complemented elements; thus studying E is closely related to studying E , the quotient of E modulo the ideal of finite sets. (From now on, the superscript will always denote that we are wor...