Positive Enumerable Functors

Csima, Barbara F.; Rossegger, Dino; Yu, Daniel

doi:10.1007/978-3-030-80049-9_38

Cited by 1 publication

(4 citation statements)

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“…The authors in [5] showed that when comparing classes of structures with respect to enumeration reducibility, positive enumerable functors are the correct effectivization of functors to consider as they preserve enumeration degree spectra.…”

Section: Discussionmentioning

confidence: 99%

“…When comparing structures with respect to their enumerations it is natural to want to use only positive information. Csima, Rossegger, and Yu [5] introduced the notion of a positive enumerable functor which uses only the positive diagrams of structures. Reductions based on this notion preserve desired properties such as enumeration degree spectra of structures.…”

Section: Functorsmentioning

confidence: 99%

“…In order to preserve enumeration degree spectra of structures we need the relationship between the two isomorphism classes to be even stronger. In [5] positive enumerable bi-transformability was introduced and it was shown that two positive enumerable bi-transformable structures have the same enumeration degree spectra. The next definition is the same as positive enumerable bi-transformability.…”

Section: F Phq Gphqmentioning

confidence: 99%

“…We show that a structure A is positively interpretable in a structure B if and only if there is a positive enumerable functor from the isomorphism class of B to the isomorphism class of A. Positive enumerable functors and related effectivizations of functors were studied by Csima, Rossegger and Yu [5]. Positive interpretability allows for a useful strengthening that preserves most properties of a structure, positive bi-interpretability.…”

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Relations enumerable from positive information

Csima¹,

MacLean²,

Rossegger³

2022

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We study countable structures from the viewpoint of enumeration reducibility. Since enumeration reducibility is based on only positive information, in this setting it is natural to consider structures given by their positive atomic diagram -the computable join of all relations of the structure. Fixing a structure A, a natural class of relations in this setting are the relations R such that R Â is enumeration reducible to the positive atomic diagram of Â for every Â -A -the relatively intrinsically positively enumerable (r.i.p.e.) relations. We show that the r.i.p.e. relations are exactly the relations that are definable by Σ p 1 formulas, a subclass of the infinitary Σ 0 1 formulas. We then introduce a new natural notion of the jump of a structure and study its interaction with other notions of jumps. At last we show that positively enumerable functors, a notion studied by Csima, Rossegger, and Yu, are equivalent to a notion of interpretability using Σ p 1 formulas.

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Section: Discussionmentioning

confidence: 99%

Section: Functorsmentioning

confidence: 99%

Section: F Phq Gphqmentioning

confidence: 99%

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confidence: 99%

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