Degrees of bi-embeddable categoricity of equivalence structures

Abstract

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) Δ⁰α 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 Δ⁰α bi-embeddable categoricity and relative Δ⁰α 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.