Computability Theoretic Properties of Injection Structures

We study computability theoretic properties of computable injection structures and the complexity of isomorphisms between these structures. We prove that a computable injection structure is computably categorical if and only if it has finitely many infinite orbits. We also prove that a computable injection structure is ∆2 categorical if and only if it has finitely many orbits of type ω or finitely many orbits of type Z. Furthermore, every computably categorical injection structure is relatively computably categorical, and every ∆2 categorical injection structure is relatively ∆ 0 2 categorical. We investigate analogues of these results for Σ1, Π 0 1, and n-c.e. injection structures. We study the complexity of the set of elements with orbits of a given type in computable injection structures. For example, we show that for every c.e. Turing degree b, there is a computable injection structure A in which the set of all elements with finite orbits has degree b and, for every Σ2 Turing degree c, there is a computable injection structure B in which the set of elements with orbits of type ω has degree c. We also study various index set results for infinite computable injection structures. For example, we show that the index set of infinite computably categorical injection structures is a Σ3 complete set and that the index set of infinite ∆2 categorical injection structure is a Σ 0 4 complete set. We also explore the connection between the complexity of the character and the first-order theory of computable injection structures. We show that for an injection structure with a computable character, there is a decidable structure isomorphic to it. However, there are computably categorical injection structures with undecidable theories.