On Dark Computably Enumerable Equivalence Relations
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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 .
Keywordsequivalence relation computably enumerable equivalence relation computable reducibility weakly precomplete equivalence relation computably enumerable order lo-reducibility
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- 1.Mal’tsev A. I., “Sets with complete numberings,” Algebra and Logic, vol. 2, no. 2, 353–378 (1963).Google Scholar
- 2.Ershov Yu. L., The Theory of Enumerations [Russian], Nauka, Moscow (1977).Google Scholar
- 11.Andrews U., Badaev S., and Sorbi A., “A survey on universal computably enumerable equivalence relations,” in: Computability and Complexity, Springer-Verlag, Cham, 2016, 418–451 (Lect. Notes Comput. Sci.; vol. 10010).Google Scholar
- 14.Ershov Yu. L. and Goncharov S. S., Constructive Models, ser. Siberian School of Algebra and Logic, Kluwer Academic/ Plenum Publishers, New York, etc. (2000).Google Scholar