On ordinals realized by c.e. equivalence relations

Following the approach of Gavryushkin, Khoussainov, and Stephan, we study countable linear orders realized by computably enumerable equivalence relations (ceers). A ceer E realizes a linear order L if there exists a computably enumerable binary relation ⊴ respecting E such that the induced quotient structure ( N / E ; ⊴ ) is isomorphic to L . The known results in this direction show that there is a highly nontrivial interplay between the computability-theoretic properties of a ceer E and the isomorphism types of orders L realizable by E . We obtain a complete characterization of well-orders L realizable by a given ceer E for the case when E realizes some ordinal α < ω ω . In particular, our characterization implies that for any ceer E with infinitely many equivalence classes, either the family of all well-orders realizable by E is contained inside the interval [ ω n ⋅ k ; ω n ⋅ ( k + 1 ) ) for some non-zero n , k < ω , or this family is cofinal in the computable ordinals. The proof of our result develops methods of fine-tuning the behavior of limit points of an E -realizable order L via the computability-theoretic properties of E .