Degrees of bi-embeddable categoricity of equivalence structures

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) Δα0 bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of Δα0 bi-embeddable categoricity and relative Δα0 bi-embeddable categoricity coincide for equivalence structures for α= 1 , 2 , 3. We also prove that computable equivalence structures have degree of bi-embeddable categoricity 0, 0′, or 0′ ′. We furthermore obtain results on the index set complexity of computable equivalence structure with respect to bi-embeddability.