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Intervals of the Lattice of Computably Enumerable Sets and Effective Boolean Algebras
, 1997
"... We prove that each interval of the lattice E of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [9]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree s ..."
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We prove that each interval of the lattice E of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [9]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree structures from complexity theory. We also answer a question left open in [6] by giving an example of a nondefinable subclass of E which has an arithmetical index set and is invariant under automorphisms. 1 Introduction Intervals play an important role in the study of the lattice E of computably enumerable (c.e.) sets under inclusion. Several interesting properties of a c.e. set can be given alternative definitions in terms of the structure of L(A), the lattice of c.e. supersets of A. For instance, hyperhypersimplicity of a coinfinite c.e. set A is equivalent to L(A) being a boolean algebra, and A is rmaximal if and only if L(A) has no nontrivial complemented elements. A further typ...
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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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.
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 "tower ..."
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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...
965 AGENDA
"... this article, we retrace the history of computability theory since 1965 in relation to the questions raised by Rogers, and try to shed a little more light on those for which solutions have yet to appear ..."
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this article, we retrace the history of computability theory since 1965 in relation to the questions raised by Rogers, and try to shed a little more light on those for which solutions have yet to appear