Computability-Theoretic Properties of Injection Structures

We study computability-theoretic properties of computable injection structures and the complexity of isomorphisms between these structures. It is proved that a computable injection structure is computably categorical iff it has finitely many infinite orbits. Again, a computable injection structure is $\Delta_2^0$ -categorical iff 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 $\Delta_2^0$ - categorical injection structure is relatively $\Delta_2^0$ -categorical. Analogs of these results are investigated for $\Sigma_1^0$ , $\Pi_1^0$ , 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, it is proved 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 $\Sigma_2^0$ 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 have various index set results for infinite computable injection structures. For example, the index set of infinite computably categorical injection structures is a $\Sigma_3^0$ -complete set, and the index set of infinite $\Delta_2^0$ -categorical injection structures is a $\Sigma_4^0$ -complete set. We explore the connection between the complexity of the character and the first-order theory of a computable injection structure. It is shown 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.