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  1. Torsion-Free Abelian Groups with Optimal Scott Families.Alexander G. Melnikov - 2018 - Journal of Mathematical Logic 18 (1):1850002.
    We prove that for any computable successor ordinal of the form α = δ + 2k there exists computable torsion-free abelian group that is relatively Δα0 -categorical and not Δα−10 -categorical. Equivalently, for any such α there exists a computable TFAG whose initial segments are uniformly described by Σαc infinitary computable formulae up to automorphism, and there is no syntactically simpler family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples (...)
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  • Computable Categoricity and the Ershov Hierarchy.Bakhadyr Khoussainov, Frank Stephan & Yue Yang - 2008 - Annals of Pure and Applied Logic 156 (1):86-95.
    In this paper, the notions of Fα-categorical and Gα-categorical structures are introduced by choosing the isomorphism such that the function itself or its graph sits on the α-th level of the Ershov hierarchy, respectively. Separations obtained by natural graphs which are the disjoint unions of countably many finite graphs. Furthermore, for size-bounded graphs, an easy criterion is given to say when it is computable-categorical and when it is only G2-categorical; in the latter case it is not Fα-categorical for any recursive (...)
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  • Automorphisms of Η -Like Computable Linear Orderings and Kierstead's Conjecture.Charles M. Harris, Kyung Il Lee & S. Barry Cooper - 2016 - Mathematical Logic Quarterly 62 (6):481-506.
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  • Computability-Theoretic Categoricity and Scott Families.Ekaterina Fokina, Valentina Harizanov & Daniel Turetsky - 2019 - Annals of Pure and Applied Logic 170 (6):699-717.
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  • On Δ 2 0 -Categoricity of Equivalence Relations.Rod Downey, Alexander G. Melnikov & Keng Meng Ng - 2015 - Annals of Pure and Applied Logic 166 (9):851-880.
  • Abelian P-Groups and the Halting Problem.Rodney Downey, Alexander G. Melnikov & Keng Meng Ng - 2016 - Annals of Pure and Applied Logic 167 (11):1123-1138.
  • Effective Categoricity of Abelian P -Groups.Wesley Calvert, Douglas Cenzer, Valentina S. Harizanov & Andrei Morozov - 2009 - Annals of Pure and Applied Logic 159 (1-2):187-197.
    We investigate effective categoricity of computable Abelian p-groups . We prove that all computably categorical Abelian p-groups are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. We investigate which computable Abelian p-groups are categorical and relatively categorical.
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  • Effective Categoricity of Equivalence Structures.Wesley Calvert, Douglas Cenzer, Valentina Harizanov & Andrei Morozov - 2006 - Annals of Pure and Applied Logic 141 (1):61-78.
    We investigate effective categoricity of computable equivalence structures . We show that is computably categorical if and only if has only finitely many finite equivalence classes, or has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k. We also prove that all computably categorical structures are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. Since all computable equivalence structures are relatively categorical, (...)
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  • The Complexity of Scott Sentences of Scattered Linear Orders.Rachael Alvir & Dino Rossegger - 2020 - Journal of Symbolic Logic 85 (3):1079-1101.
    We calculate the complexity of Scott sentences of scattered linear orders. Given a countable scattered linear order L of Hausdorff rank $\alpha $ we show that it has a ${d\text {-}\Sigma _{2\alpha +1}}$ Scott sentence. It follows from results of Ash [2] that for every countable $\alpha $ there is a linear order whose optimal Scott sentence has this complexity. Therefore, our bounds are tight. We furthermore show that every Hausdorff rank 1 linear order has an optimal ${\Pi ^{\mathrm {c}}_{3}}$ (...)
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