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  1. Asher M. Kach, Karen Lange & Reed Solomon (2013). Degrees of Orders on Torsion-Free Abelian Groups. Annals of Pure and Applied Logic 164 (7-8):822-836.
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  2. Brooke M. Andersen, Asher M. Kach, Alexander G. Melnikov & Reed Solomon (2012). Jump Degrees of Torsion-Free Abelian Groups. Journal of Symbolic Logic 77 (4):1067-1100.
    We show, for each computable ordinal α and degree $\alpha > {0^{\left( \alpha \right)}}$, the existence of a torsion-free abelian group with proper α th jump degree α.
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  3. Rodney G. Downey, Sergei S. Goncharov, Asher M. Kach, Julia F. Knight, Oleg V. Kudinov, Alexander G. Melnikov & Daniel Turetsky (2010). Decidability and Computability of Certain Torsion-Free Abelian Groups. Notre Dame Journal of Formal Logic 51 (1):85-96.
    We study completely decomposable torsion-free abelian groups of the form $\mathcal{G}_S := \oplus_{n \in S} \mathbb{Q}_{p_n}$ for sets $S \subseteq \omega$. We show that $\mathcal{G}_S$has a decidable copy if and only if S is $\Sigma^0_2$and has a computable copy if and only if S is $\Sigma^0_3$.
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  4. Rodney G. Downey & Asher M. Kach (2010). Euclidean Functions of Computable Euclidean Domains. Notre Dame Journal of Formal Logic 52 (2):163-172.
    We study the complexity of (finitely-valued and transfinitely-valued) Euclidean functions for computable Euclidean domains. We examine both the complexity of the minimal Euclidean function and any Euclidean function. Additionally, we draw some conclusions about the proof-theoretical strength of minimal Euclidean functions in terms of reverse mathematics.
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  5. Asher M. Kach, Oscar Levin & Reed Solomon (2010). Embeddings of Computable Structures. Notre Dame Journal of Formal Logic 51 (1):55-68.
    We study what the existence of a classical embedding between computable structures implies about the existence of computable embeddings. In particular, we consider the effect of fixing and varying the computable presentations of the computable structures.
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  6. Asher M. Kach & Daniel Turetsky (2010). Limitwise Monotonic Functions, Sets, and Degrees on Computable Domains. Journal of Symbolic Logic 75 (1):131-154.
    We extend the notion of limitwise monotonic functions to include arbitrary computable domains. We then study which sets and degrees are support increasing (support strictly increasing) limitwise monotonic on various computable domains. As applications, we provide a characterization of the sets S with computable increasing η-representations using support increasing limitwise monotonic sets on ${\Bbb Q}$ and note relationships between the class of order-computable sets and the class of support increasing (support strictly increasing) limitwise monotonic sets on certain domains.
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  7. Asher M. Kach (2008). Computable Shuffle Sums of Ordinals. Archive for Mathematical Logic 47 (3):211-219.
    The main result is that for sets ${S \subseteq \omega + 1}$ , the following are equivalent: The shuffle sum σ(S) is computable.The set S is a limit infimum set, i.e., there is a total computable function g(x, t) such that ${f(x) = \lim inf_t g(x, t)}$ enumerates S.The set S is a limitwise monotonic set relative to 0′, i.e., there is a total 0′-computable function ${\tilde{g}(x, t)}$ satisfying ${\tilde{g}(x, t) \leq \tilde{g}(x, t+1)}$ such that ${{\tilde{f}(x) = \lim_t \tilde{g}(x, t)}}$ (...)
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