Weakly precomplete equivalence relations in the Ershov hierarchy

Bazhenov, N. A.; Kalmurzaev, B. S.

doi:10.33048/alglog.2019.58.301

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2019

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“…The recent works [9,10,11] started systematic investigations of c-degrees for ∆ 0 2 equivalence relations. We note that computable reducibility has been also studied for higher levels of the hyperarithmetical hierarchy, but these studies were largely focused on complete equivalence relations.…”

Section: Introductionmentioning

confidence: 99%

Minimal Equivalence Relations in Hyperarithmetical and Analytical Hierarchies

Bazhenov¹,

Mustafa²,

Mauro³

et al. 2019

Preprint

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A standard tool for classifying the complexity of equivalence relations on ω is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce minimal degrees with respect to computable reducibility. Let Γ be one of the following classes: Σ 0 α , Π 0 α , Σ 1 n , or Π 1 n , where α ě 2 is a computable ordinal and n is a non-zero natural number. We prove that there are infinitely many pairwise incomparable minimal equivalence relations that are properly in Γ.

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Section: Introductionmentioning

confidence: 99%