On Dark Computably Enumerable Equivalence Relations

N. A. Bazhenov, B. S. Kalmurzaev

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)22-30
Number of pages9
JournalSiberian Mathematical Journal
Issue number1
Publication statusPublished - 1 Jan 2018


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


Dive into the research topics of 'On Dark Computably Enumerable Equivalence Relations'. Together they form a unique fingerprint.

Cite this