Positive Structures

Selivanov, Victor L.

doi:10.1007/978-1-4615-0755-0_14

Cited by 23 publications

(9 citation statements)

References 46 publications

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“…Boolean algebras in the sense of [18,26]. See also [23,11,12,9] for an approach to c.e structures similar to the one taken in this paper.…”

Section: Definitionmentioning

confidence: 98%

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Effective Inseparability, Lattices, and Preordering Relations

Andrews¹,

Sorbi²

2019

The Review of Symbolic Logic

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We study effectively inseparable (e.i.) pre-lattices (i.e. structures of the form L " xω,^, _, 0, 1, ďLy where ω denotes the set of natural numbers and the following hold:^, _ are binary computable operations; ďL is a c.e. pre-ordering relation, with 0 ďL x ďL 1 for every x; the equivalence relation "L originated by ďL is a congruence on L such that the corresponding quotient structure is a non-trivial bounded lattice; the "L-equivalence classes of 0 and 1 form an effectively inseparable pair), and show (Theorem 30, solving a problem in [17]), that if L is an e.i. pre-lattice then ďL is universal with respect to all c.e. pre-ordering relations, i.e. for every c.e. pre-ordering relation R there exists a computable function f such that, for all x, y, x R y if and only if f pxq ďL f pyq; in fact (Corollary 43) ďL is locally universal, i.e. for every pair a ăL b and every c.e. pre-ordering relation R one can find a reducing function f from R to ďL such that the range of f is contained in the interval tx : a ďL x ďL bu. Also (Theorem 47) ďL is uniformly dense, i.e. there exists a computable function f such that for every a, b if a ăL b then a ăL f pa, bq ăL b, and if a "L a 1 and b "L b 1 then f pa, bq "L f pa 1 , b 1 q. Some consequences and applications of these results are discussed: in particular (Corollary 55) for n ě 1 the c.e. pre-ordering relation on Σn sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson's Q or R is locally universal and uniformly dense; and (Corollary 56) the c.e. pre-ordering relation of provable implication of Heyting Arithmetic is locally universal and uniformly dense.2010 Mathematics Subject Classification. 03D25.

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“…Boolean algebras in the sense of [18,26]. See also [23,11,12,9] for an approach to c.e structures similar to the one taken in this paper.…”

Section: Definitionmentioning

confidence: 98%

“…Boolean algebras in the sense of (Nerode & Remmel, 1985;Shavrukov, 2010). See also (Fokina, Khoussainov, Semukhin, & Turetsky, 2016;Gavryushkin, Jain, Khoussainov, & Stephan, 2014;Gavryushkin, Khoussainov, & Stephan, 2016;Selivanov, 2003) for an approach to c.e structures similar to the one taken in this article.…”

mentioning

confidence: 99%

Effective Inseparability, Lattices, and Preordering Relations

Andrews¹,

Sorbi²

2019

The Review of Symbolic Logic

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“…The study of quotient presentations dates back to the very beginning of computable model theory (see, e.g., the work of Metakides and Nerode on c.e. vector spaces [20]) and it gave rise to a rich research program; the interested reader is referred to [21,22,23]. However, it should be clear that in this paper we do not deal with quotients, but rather with the (pre-)structure themselves.…”

Section: Warning About Our Terminologymentioning

confidence: 99%

Comparing the isomorphism types of equivalence structures and preorders

Bazhenov,

Mauro

2020

Preprint

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A general theme of computable structure theory is to investigate when structures have copies of a given complexity Γ. We discuss such problem for the case of equivalence structures and preorders. We show that there is a Π 0 1 equivalence structure with no Σ 0 1 copy, and in fact that the isomorphism types realized by the Π 0 1 equivalence structures coincide with those realized by the ∆ 0 2 equivalence structures. We also construct a Σ 0 1 preorder with no Π 0 1 copy.2010 Mathematics Subject Classification. 03D45.

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“…presentations of structures, as is shown for instance in [ 13 , 18 ] (for a nice survey about c.e. structures, see [ 23 ]).…”

Section: Introductionmentioning

confidence: 99%

Classifying equivalence relations in the Ershov hierarchy

et al. 2020

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Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility ďc. This gives rise to a rich degree-structure. In this paper, we lift the study of c-degrees to the ∆ 0 2 case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree-structure induced by ďc on the Σ´1 a Π´1 a equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.

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Positive Structures

Cited by 23 publications

References 46 publications

Effective Inseparability, Lattices, and Preordering Relations

Effective Inseparability, Lattices, and Preordering Relations

Comparing the isomorphism types of equivalence structures and preorders

Classifying equivalence relations in the Ershov hierarchy

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