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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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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.
On the nbackandforth types of Boolean algebras
 In preparation
"... Abstract. The objective of this paper is to uncover the structure of the backandforth equivalence classes at the finite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the backandforth equivalence classes of a Boolean al ..."
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Abstract. The objective of this paper is to uncover the structure of the backandforth equivalence classes at the finite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the backandforth equivalence classes of a Boolean algebra for finite levels. This result has implications for characterizing the relatively intrinsically Σ 0 n relations of Boolean algebras as existential formulas over a finite set of relations. 1.
BOOLEAN ALGEBRA APPROXIMATIONS
"... Abstract. Knight and Stob proved that every low4 Boolean algebra is 0 (6)isomorphic to a computable one. Furthermore, for n = 1, 2, 3, 4, every lown Boolean algebra is 0 (n+2)isomorphic to a computable one. We show that this is not true for n = 5: there is a low5 Boolean algebra that is not 0 (7) ..."
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Abstract. Knight and Stob proved that every low4 Boolean algebra is 0 (6)isomorphic to a computable one. Furthermore, for n = 1, 2, 3, 4, every lown Boolean algebra is 0 (n+2)isomorphic to a computable one. We show that this is not true for n = 5: there is a low5 Boolean algebra that is not 0 (7)isomorphic to any computable Boolean algebra. It is worth remarking that, because of the machinery developed, the proof uses at most a 0 ′ ′priority argument. The technique used to construct this Boolean algebra is new and might be useful in other applications, such as to solve the lown Boolean algebra problem either positively or negatively. 1.
DEPTH ZERO BOOLEAN ALGEBRAS
, 2010
"... Abstract. We study the class of depth zero Boolean algebras, both from a classical viewpoint and an effective viewpoint. In particular, we provide an algebraic characterization, constructing an explicit measure for each depth zero Boolean algebra and demonstrating there are no others, and an effecti ..."
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Abstract. We study the class of depth zero Boolean algebras, both from a classical viewpoint and an effective viewpoint. In particular, we provide an algebraic characterization, constructing an explicit measure for each depth zero Boolean algebra and demonstrating there are no others, and an effective characterization, providing a necessary and sufficient condition for a depth zero Boolean algebra of rank at most ω to have a computable presentation. 1.
NOTES ON THE JUMP OF A STRUCTURES
"... Abstract. We introduce the notions of a complete set of computably infinitary Π 0 n relations on a structure, of the jump of a structure, and of admitting nth jump inversion. ..."
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Abstract. We introduce the notions of a complete set of computably infinitary Π 0 n relations on a structure, of the jump of a structure, and of admitting nth jump inversion.
CUTS OF LINEAR ORDERS
"... Abstract. We study the connection between the number of ascending and descending cuts of a linear order, its classical size, and its effective complexity (how much [how little] information can be encoded into it). 1. ..."
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Abstract. We study the connection between the number of ascending and descending cuts of a linear order, its classical size, and its effective complexity (how much [how little] information can be encoded into it). 1.
COUNTING THE BACKANDFORTH TYPES
"... Abstract. Given a class of structures K and n ∈ ω, we study the dichotomy between there being countably many nbackandforth equivalence classes and there being continuum many. In the latter case we show that, relative to some oracle, every set can be weakly coded in the (n − 1)st jump of some stru ..."
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Abstract. Given a class of structures K and n ∈ ω, we study the dichotomy between there being countably many nbackandforth equivalence classes and there being continuum many. In the latter case we show that, relative to some oracle, every set can be weakly coded in the (n − 1)st jump of some structure in K. In the former case we show that there is a countable set of infinitary Πn relations that captures all of the Πn information about the structures in K. In most cases where there are countably many nbackandforth equivalence classes, there is a computable description of them. We will show how to use this computable description to get a complete set of computably infinitary Πn formulas. This will allow us to completely characterize the relatively intrinsically Σ 0 n+1 relations in the computable structures of K, and to prove that no Turing degree can be coded by the (n − 1)st jump of any structure in K unless that degree is already below 0 (n−1). 1.
Russian Academy of Sciences, Siberian Branch
, 2010
"... We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of nonlown degrees. We then repeat these questions specifically for linear orders and for relations on the ..."
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We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of nonlown degrees. We then repeat these questions specifically for linear orders and for relations on the computable dense linear order Q. New results include realizations of the set of nonlown Turing degrees as the spectrum of a relation on Q for all n ≥ 1, and a realization of the set of all nonlown Turing degrees as the spectrum of a linear order whenever n ≥ 2. The state of current knowledge is summarized in a table in the concluding section.