Abstract
A standard tool for classifying the complexity of equivalence relations on \(\omega\) 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 \(\Gamma\) be one of the following classes: \(\Sigma^{0}_{\alpha}\), \(\Pi^{0}_{\alpha}\), \(\Sigma^{1}_{n}\), or \(\Pi^{1}_{n}\), where \(\alpha\geq 2\) 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 \(\Gamma\).
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ACKNOWLEDGMENTS
Part of the research contained in this paper was carried out while Bazhenov, San Mauro, and Yamaleev were visiting the Department of Mathematics of Nazarbayev University, Nur-Sultan. The authors wish to thank Nazarbayev University for its hospitality.
Funding
The work was supported by Nazarbayev University Faculty Development Competitive Research Grants N090118FD5342. Bazhenov was supported by the grant of the President of the Russian Federation (No. MK-1214.2019.1). San Mauro was supported by the Austrian Science Fund FWF, project M 2461. Yamaleev was supported by the Russian Science Foundation, project No. 18-11-00028.
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Bazhenov, N.A., Mustafa, M., San Mauro, L. et al. Minimal Equivalence Relations in Hyperarithmetical and Analytical Hierarchies. Lobachevskii J Math 41, 145–150 (2020). https://doi.org/10.1134/S199508022002002X
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DOI: https://doi.org/10.1134/S199508022002002X