On dark computably enumerable equivalence relations

We study computably enumerable (c.e.) relations on the set of naturals. A binary relation $R$ on $\omega$ is computably reducible to a relation $S$ (which is denoted by $R\leq_cS$) 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 $\leq_c$ 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 $E\leq_cF$. As a consequence of this result, we construct an infinite increasing $\leq_c$-chain of weakly precomplete dark c.e. equivalence relations. We also show the existence of a universal c.e. linear order with respect to $\leq_c$.