Effective categoricity of equivalence structures

We investigate effective categoricity of computable equivalence structures A . We show that A is computably categorical if and only if A has only finitely many finite equivalence classes, or A has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k . We also prove that all computably categorical structures are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. Since all computable equivalence structures are relatively Δ 3 0 categorical, we further investigate when they are Δ 2 0 categorical. We also obtain results on the index sets of computable equivalence structures.