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.
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Translated from Algebra i Logika, Vol. 53, No. 1, pp. 60-108, January-February, 2014.
∗Supported by NSF, grant DMS-652372.
∗∗Supported by NSF, grant DMS-1202328.
∗∗∗Supported by NSF, grant DMS-0654060.
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Cenzer*, D., Harizanov**, V. & Remmel***, J.B. Computability-Theoretic Properties of Injection Structures. Algebra Logic 53, 39–69 (2014). https://doi.org/10.1007/s10469-014-9270-0
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DOI: https://doi.org/10.1007/s10469-014-9270-0