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Siberian Mathematical Journal

, Volume 59, Issue 1, pp 22–30 | Cite as

On Dark Computably Enumerable Equivalence Relations

  • N. A. Bazhenov
  • B. S. Kalmurzaev
Article
  • 22 Downloads

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 c S) 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 c F. 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 .

Keywords

equivalence relation computably enumerable equivalence relation computable reducibility weakly precomplete equivalence relation computably enumerable order lo-reducibility 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  2. 2.Al-Farabi Kazakh National UniversityAlmatyKazakhstan

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