We study the complexity of natural equivalence relations on classes of computable structures such as isomorphism and bi-embeddability. We use the notion of tc-reducibility to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ω. We also show that the bi-embeddability relation on an appropriate hyperarithmetical class of computable structures may have the same complexity as any given Σ 1 1 equivalence relation on ω. ∗The first and the second authors acknowledge the generous support of the FWF through

Abstract. We study completely decomposable torsion-free abelian groups of the form GS: = ⊕n∈SQpn for sets S ⊆ ω. We show that GS has a decidable copy if and only if S is Σ0 2 and has a computable copy if and only if S is Σ0 3. 1.

Abstract. We study effective categoricity of computable abelian groups of the form ⊕ i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is well-known that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite. We study ∆0 n-categoricity in this class of groups, for n> 1. We introduce a new algebraic concept of S-independence which is a generalization of the wellknown notion of p-independence. We develop the theory of P-independent sets. We apply these techniques to show that every homogeneous completely decomposable group is ∆0 3-categorical. We prove that a homogeneous completely decomposable group of infinite rank is ∆0 2-categorical if and only if it is isomorphic to the free module over the localization of Z by a computably enumerable set of primes P with the semi-low complement (within the set of all primes). Finally, we apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers, including the free abelian group.

Abstract. We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1-decidable computably categorical structure is relatively ∆0 2-categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also introduce and study a variation of relative computable categoricity, comparing it to both computable categoricity and relative computable categoricity and its relativizations. 1.