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  1. The Complexity of Decomposability of Computable Rings.Huishan Wu - 2023 - Notre Dame Journal of Formal Logic 64 (1):1-14.
    This article studies the complexity of decomposability of rings from the perspective of computability. Based on the equivalence between the decomposition of rings and that of the identity of rings, we propose four kinds of rings, namely, weakly decomposable rings, decomposable rings, weakly block decomposable rings, and block decomposable rings. Let R be the index set of computable rings. We study the complexity of subclasses of computable rings, showing that the index set of computable weakly decomposable rings is m-complete Σ10 (...)
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  • The computational complexity of module socles.Huishan Wu - 2022 - Annals of Pure and Applied Logic 173 (5):103089.
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  • 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 Topological Groups.K. O. H. Heer Tern, Alexander G. Melnikov & N. G. Keng Meng - forthcoming - Journal of Symbolic Logic:1-33.
    We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare our results with similar results in computable topology.
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  • Computability of polish spaces up to homeomorphism.Matthew Harrison-Trainor, Alexander Melnikov & Keng Meng Ng - 2020 - Journal of Symbolic Logic 85 (4):1664-1686.
    We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a $\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal $\alpha $, an effectively closed set not homeomorphic to any $0^{}$-computable Polish space; this answers a question of Nies. (...)
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  • Primitive recursive reverse mathematics.Nikolay Bazhenov, Marta Fiori-Carones, Lu Liu & Alexander Melnikov - 2024 - Annals of Pure and Applied Logic 175 (1):103354.
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