On Dark Computably Enumerable Equivalence Relations

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 .