Notre Dame Journal of Formal Logic

On the Degree Structure of Equivalence Relations Under Computable Reducibility

Keng Meng Ng and Hongyuan Yu

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Abstract

We study the degree structure of the ω-c.e., n-c.e., and Π10 equivalence relations under the computable many-one reducibility. In particular, we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-computably enumerable equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e., and Π10 equivalence relations. We prove that for all the degree classes considered, upward density holds and downward density fails.

Article information

Source
Notre Dame J. Formal Logic, Volume 60, Number 4 (2019), 733-761.

Dates
Received: 5 November 2017
Accepted: 18 July 2018
First available in Project Euclid: 12 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1568253623

Digital Object Identifier
doi:10.1215/00294527-2019-0028

Subjects
Primary: 03D28: Other Turing degree structures
Secondary: 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]

Keywords
degrees equivalence relations density effective enumerations universality

Citation

Ng, Keng Meng; Yu, Hongyuan. On the Degree Structure of Equivalence Relations Under Computable Reducibility. Notre Dame J. Formal Logic 60 (2019), no. 4, 733--761. doi:10.1215/00294527-2019-0028. https://projecteuclid.org/euclid.ndjfl/1568253623


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