On Dark Computably Enumerable Equivalence Relations
- Authors: Bazhenov N.A.1, Kalmurzaev B.S.2
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Affiliations:
- Sobolev Institute of Mathematics
- Al-Farabi Kazakh National University
- Issue: Vol 59, No 1 (2018)
- Pages: 22-30
- Section: Article
- URL: https://journals.rcsi.science/0037-4466/article/view/171638
- DOI: https://doi.org/10.1134/S0037446618010032
- ID: 171638
Cite item
Abstract
We study computably enumerable (c.e.) relations on the set of naturals. A binary relation R on ω is computably reducible to a relation S (which is denoted by R ≤cS) if there exists a computable function f(x) such that the conditions (xRy) and (f(x)Sf(y)) are equivalent for all x and y. An equivalence relation E is called dark if it is incomparable with respect to ≤c with the identity equivalence relation. We prove that, for every dark c.e. equivalence relation E there exists a weakly precomplete dark c.e. relation F such that E ≤cF. As a consequence of this result, we construct an infinite increasing ≤c-chain of weakly precomplete dark c.e. equivalence relations. We also show the existence of a universal c.e. linear order with respect to ≤c.
About the authors
N. A. Bazhenov
Sobolev Institute of Mathematics
Author for correspondence.
Email: bazhenov@math.nsc.ru
Russian Federation, Novosibirsk
B. S. Kalmurzaev
Al-Farabi Kazakh National University
Email: bazhenov@math.nsc.ru
Kazakhstan, Almaty