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14
On Presentations of Algebraic Structures
 in Complexity, Logic and Recursion Theory
, 1995
"... This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicat ..."
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Cited by 17 (6 self)
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This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995
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 16 (5 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.
Effective presentability of Boolean algebras of CantorBendixson rank 1
 Journal of Symbolic Logic
, 1999
"... We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B/I is of CantorBendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though F ..."
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Cited by 6 (6 self)
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We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B/I is of CantorBendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite CantorBendixson rank.
Questions in Computable Algebra and Combinatorics
, 1999
"... this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these ..."
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Cited by 5 (0 self)
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this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25
Effectively dense Boolean algebras and their applications
"... A computably enumerable boolean algebra B is effectively dense if for each x 2 B we can effectively determine an F (x) x such that x 6= 0 implies 0 ! F (x) ! x. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a boolean algebra. A ..."
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Cited by 4 (3 self)
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A computably enumerable boolean algebra B is effectively dense if for each x 2 B we can effectively determine an F (x) x such that x 6= 0 implies 0 ! F (x) ! x. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of E (the lattice of computably enumerable sets under inclusion) which are not boolean algebras. We derive a similar result for theories of certain initial segments "low down" of subrecursive degree structures. 1 Introduction We describe a uniform method to interpret Th(N; +; \Theta) in the theories of a wide variety of seemingly wellbehaved structures. These structures stem from formal logic, complexity theory and computability theory. In many cases, they are closely related to dense distributive lattices. The results can be summarized by saying that, in spite of the structure's apparen...
π0 1 classes and orderable groups
 Annals of Pure and Applied Logic
"... It is known that the spaces of orders on orderable computable fields can represent all Π0 1 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent Π0 1 classes in even a weak manner. Next, we consider presentations of orde ..."
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Cited by 4 (0 self)
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It is known that the spaces of orders on orderable computable fields can represent all Π0 1 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent Π0 1 classes in even a weak manner. Next, we consider presentations of ordered abelian groups, and we show that there is a computable ordered abelian group for which no computable presentation admits a computable set of representatives for its Archimedean classes. 1
Slender classes
, 2006
"... Abstract. A Π 0 1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩ C. Cholak, Coles, Downey and Herrmann [7] proved that a Π 0 1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean al ..."
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Cited by 2 (1 self)
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Abstract. A Π 0 1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩ C. Cholak, Coles, Downey and Herrmann [7] proved that a Π 0 1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is the free Boolean algebra, then all such think classes are automorphic in the lattice of Π 0 1 classes under inclusion. From this it follows that if the boolean algebra has a finite number n of atoms then the resulting classes are all automorphic. We prove a conjecture of Cholak and Downey [8] by showing that this is the only time the Boolean algebra determines the automorphism type of a thin class. 1.
Computability, Definability and Algebraic Structures
, 1999
"... In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set ..."
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Cited by 2 (1 self)
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In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set
DECIDABILITY AND COMPUTABILITY OF CERTAIN TORSIONFREE ABELIAN GROUPS
"... Abstract. We study completely decomposable torsionfree abelian groups of the form GS: = ⊕n∈SQpn for sets S ⊆ ω. We show that GS has a decidable copy if and only if S is Σ0 2 and has a computable copy if and only if S is Σ0 3. 1. ..."
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Abstract. We study completely decomposable torsionfree abelian groups of the form GS: = ⊕n∈SQpn for sets S ⊆ ω. We show that GS has a decidable copy if and only if S is Σ0 2 and has a computable copy if and only if S is Σ0 3. 1.
Computably enumerable algebras, their expansions
 and isomorphisms, The International Journal of Algebra and Computation, accepted
"... Abstract. Computably enumerable algebras are the ones whose positive atomic diagrams are computably enumerable. Computable algebras are the ones whose atomic diagrams are computable. In this paper we investigate computably enumerable algebras and provide several algebraic and computable theoretic di ..."
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Cited by 1 (1 self)
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Abstract. Computably enumerable algebras are the ones whose positive atomic diagrams are computably enumerable. Computable algebras are the ones whose atomic diagrams are computable. In this paper we investigate computably enumerable algebras and provide several algebraic and computable theoretic distinctions of these algebras from the class of computable algebras. We give a characterization of computably enumerable but not computable algebras in terms of congruences and effective conjunctions of Π 0 1sentences. Our characterization, for example, shows that computable conjunctions of negative atomic formulas true in a given c.e. algebra can be preserved in infinitely many of its homomorphic images. We also study questions on how expansions of algebras by finitely many new functions affect computable isomorphism types. In particular, we construct a c.e. algebra with unique computable isomorphism type but which has no finitely generated c.e. expansion. 1