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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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  • New Degree Spectra of Abelian Groups.Alexander G. Melnikov - 2017 - Notre Dame Journal of Formal Logic 58 (4):507-525.
    We show that for every computable ordinal of the form β=δ+2n+1>1, where δ is zero or a limit ordinal and n∈ω, there exists a torsion-free abelian group having an X-computable copy if and only if X is nonlowβ.
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  • Computable Abelian Groups.Alexander G. Melnikov - 2014 - Bulletin of Symbolic Logic 20 (3):315-356,.
    We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem.
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