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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
The Continuity of Cupping to 0
 Annals of Pure and Applied Logic
, 1993
"... It is shown that, if a, b are recursively enumerable degrees such that 0 ! a ! 0 0 and a [ b = 0 0 , then there exists a recursively enumerable degree c such that c ! a and c [ b = 0 0 . 1 Introduction By analogy with the notion of major subset in the context of the lattice of r.e. sets, the ..."
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It is shown that, if a, b are recursively enumerable degrees such that 0 ! a ! 0 0 and a [ b = 0 0 , then there exists a recursively enumerable degree c such that c ! a and c [ b = 0 0 . 1 Introduction By analogy with the notion of major subset in the context of the lattice of r.e. sets, the r.e. degree c is called a major subdegree of the r.e. degree a if c ! a and for every r.e. degree b a [ b = 0 0 ) c [ b = 0 0 : This paper represents modest progress towards answering the question: Does every r.e. degree which is neither 0 nor 0 0 have a major subdegree? This question was first posed by the second author in 1967, although it does not seem to have appeared in print. In the 70's and 80's efforts were made to answer the question but bore little fruit. In this paper we prove: The second author was supported by NSERC (Canada) Grant A3040, and the third author by National Science Foundation Grant DMS 8807389. The third author presented the results in this paper and tho...
Extension of embeddings in the recursively enumerable degrees
"... The extension of embeddings problem for the recursively enumerable degrees R = (R;!; 0; 0 0) asks for given finite partially ordered sets P ` Q with least and greatest elements, whether every embedding of P into R can be extended to an embedding of Q into R. Many of the landmark theorems giving an a ..."
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The extension of embeddings problem for the recursively enumerable degrees R = (R;!; 0; 0 0) asks for given finite partially ordered sets P ` Q with least and greatest elements, whether every embedding of P into R can be extended to an embedding of Q into R. Many of the landmark theorems giving an algebraic insight into R assert either extension or nonextension of embeddings. We extend, strengthen, and unify these results and their proofs to produce complete and complementary criteria and techniques to analyze instances of extension and nonextension. We conclude that the full extension of embeddings problem is decidable.
Structural Properties of D.C.E. Degrees and Presentations of C.E. Reals
"... To my wife Caixia and my daughter Jiahui Abstract In this thesis, we are mainly concerned with the structural properties of the d.c.e. degrees and the distribution of the simple reals among the c.e. degrees. In chapters 2 and 3, we study the relationship between the isolation phenomenon and the jump ..."
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To my wife Caixia and my daughter Jiahui Abstract In this thesis, we are mainly concerned with the structural properties of the d.c.e. degrees and the distribution of the simple reals among the c.e. degrees. In chapters 2 and 3, we study the relationship between the isolation phenomenon and the jump operator. We prove in chapter 2 that there is a high d.c.e. degree d isolated by a low2 degree a. We improve this result in chapter 3 by showing that the isolating degree a can be low. Chapters 4 and 5 are devoted to the study of the pseudoisolation in the d.c.e. degrees. We prove that pseudoisolated d.c.e. degrees are dense in the c.e. degrees, and that there is a high d.c.e. degree pseudoisolated by a low d.c.e. degree.
Continuity of capping in EbT
, 2007
"... A set A ⊆ ω is called computably enumerable (c.e., for short), if there is an algorithm to enumerate the elements of it. For sets A, B ⊆ ω, we say that A is bounded Turing reducible to B if there is a Turing functional, Φ say, with a computable bound of oracle query bits such that A is computed by Φ ..."
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A set A ⊆ ω is called computably enumerable (c.e., for short), if there is an algorithm to enumerate the elements of it. For sets A, B ⊆ ω, we say that A is bounded Turing reducible to B if there is a Turing functional, Φ say, with a computable bound of oracle query bits such that A is computed by Φ equipped with an oracle B, written A ≤bT B. Let EbT be the structure of the c.e. bTdegrees, the c.e. degrees under the bounded Turing reductions. In this paper we study the continuity properties in EbT. We show that for any c.e. bTdegree b � = 0, 0 ′ , there is a c.e. bTdegree a> b such that for any c.e. bTdegree x, b ∧ x = 0 if and only if a ∧ x = 0. This is the first continuity property of the c.e. bTdegrees. 1
ON DEFINABLE FILTERS IN COMPUTABLY ENUMERABLE DEGREES
"... Abstract. Based on a result of Nies on definability the upper semilattice of computably enumerable degrees (denoted by R), we find that in R filters generated by definable subsets are also definable. As applications we demonstrate two new definable filters and study their supremum. Finally we demon ..."
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Abstract. Based on a result of Nies on definability the upper semilattice of computably enumerable degrees (denoted by R), we find that in R filters generated by definable subsets are also definable. As applications we demonstrate two new definable filters and study their supremum. Finally we demonstrate a counterexample. 1.