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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
Asymptotic density and computably enumerable sets
"... We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in ..."
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Cited by 9 (6 self)
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We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: (i) The degrees of such sets A are precisely the nonlow c.e. degrees. (ii) There is a c.e. set A of density 1 with no computable subset of nonzero density. (iii) There is a c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A \ B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of “computable at density r ” where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness.
EFFECTIVELY CATEGORICAL ABELIAN GROUPS
"... We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is wellknown that a homogeneous completely decomposable group is computably categorical if and only if its rank is finit ..."
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We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is wellknown that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite. We study ∆0 ncategoricity in this class of groups, for n> 1. We introduce a new algebraic concept of Sindependence which is a generalization of the wellknown notion of pindependence. We develop the theory of Pindependent sets. We apply these techniques to show that every homogeneous completely decomposable group is ∆0 3categorical. We prove that a homogeneous completely decomposable group of infinite rank is ∆0 2categorical if and only if it is isomorphic to the free module over the localization of Z by a computably enumerable set of primes P with the semilow complement (within the set of all primes). Finally, we apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers, including the free abelian group.
An Overview of the Computably Enumerable Sets
"... The purpose of this article is to summarize some of the results on the algebraic structure of the computably enumerable (c.e.) sets since 1987 when the subject was covered in Soare 1987 , particularly Chapters X, XI, and XV. We study the c.e. sets as a partial ordering under inclusion, (E; `). We do ..."
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Cited by 2 (0 self)
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The purpose of this article is to summarize some of the results on the algebraic structure of the computably enumerable (c.e.) sets since 1987 when the subject was covered in Soare 1987 , particularly Chapters X, XI, and XV. We study the c.e. sets as a partial ordering under inclusion, (E; `). We do not study the partial ordering of the c.e. degrees under Turing reducibility, although a number of the results here relate the algebraic structure of a c.e. set A to its (Turing) degree in the sense of the information content of A. We consider here various properties of E: (1) deønable properties; (2) automorphisms; (3) invariant properties; (4) decidability and undecidability results; miscellaneous results. This is not intended to be a comprehensive survey of all results in the subject since 1987, but we give a number of references in the bibliography to other results.
Effectively and noneffectively nowhere simple sets
 Mathematical Logic Quarterly 42
, 1996
"... Abstract. R. Shore proved that every recursively enumerable (r.e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice E of all r.e. sets. Nowhere simple sets were further studied by ..."
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Abstract. R. Shore proved that every recursively enumerable (r.e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice E of all r.e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r.e. sets which can be split into two (non)effectively nowhere simple sets, and r.e. sets which can be split into two r.e. nonnowhere simple sets. We show that every r.e. set is either the disjoint union of two effectivelynowheresimplesets or two noneffectively nowhere simple sets. We characterize r.e. sets whose every nontrivial splitting is into nowhere simple sets, and r.e. sets whose every nontrivial splitting is into effectively nowhere simple sets. R. Shore proved that for every effectively nowhere simple set A, thelattice L ∗ (A) is effectively isomorphic to E ∗ , and that there is a nowhere simple set A such that L ∗ (A) is not effectively isomorphic to E ∗. We prove that every nonzero r.e. Turing degree contains a noneffectively nowhere simple set A with the lattice L ∗ (A) effectively isomorphic to E ∗. 1. Introduction and
PROMPT SIMPLICITY, ARRAY COMPUTABILITY AND CUPPING
"... Abstract. We show that the class of c.e. degrees that can be joined to 0 ′ by an array computable c.e. degree properly contains the class of promptly simple degrees. 1. ..."
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Abstract. We show that the class of c.e. degrees that can be joined to 0 ′ by an array computable c.e. degree properly contains the class of promptly simple degrees. 1.
Definable properties of the computably enumerable sets
 Proceedings of the Oberwolfach Conference on Computability Theory
, 1996
"... Post 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, hsimple, and hhsimple, with the intent of finding a property guaranteeing incompleteness of A. From observations of Post 1943 and Myhill 1956, attention focused by the 1950's on properties definable in ..."
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Post 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, hsimple, and hhsimple, with the intent of finding a property guaranteeing incompleteness of A. From observations of Post 1943 and Myhill 1956, attention focused by the 1950's on properties definable in the inclusion ordering of c.e. subsets of!, namely E = (fWngn2! ; ae). In the 1950's and 1960's Tennenbaum, Martin, Yates, Sacks, Lachlan, Shoenfield and others produced a number of elegant results relating Edefinable properties of A, like maximal, hhsimple, atomless, to the information content (usually the