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Karen Lange [12]Karen M. Lange [1]
  1. Degrees of Orders on Torsion-Free Abelian Groups.Asher M. Kach, Karen Lange & Reed Solomon - 2013 - Annals of Pure and Applied Logic 164 (7-8):822-836.
    We show that if H is an effectively completely decomposable computable torsion-free abelian group, then there is a computable copy G of H such that G has computable orders but not orders of every degree.
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  2.  10
    Bounded Low and High Sets.Bernard A. Anderson, Barbara F. Csima & Karen M. Lange - 2017 - Archive for Mathematical Logic 56 (5-6):507-521.
    Anderson and Csima :245–264, 2014) defined a jump operator, the bounded jump, with respect to bounded Turing reducibility. They showed that the bounded jump is closely related to the Ershov hierarchy and that it satisfies an analogue of Shoenfield jump inversion. We show that there are high bounded low sets and low bounded high sets. Thus, the information coded in the bounded jump is quite different from that of the standard jump. We also consider whether the analogue of the Jump (...)
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  3.  19
    Computability of Homogeneous Models.Karen Lange & Robert I. Soare - 2007 - Notre Dame Journal of Formal Logic 48 (1):143-170.
    In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of homogeneous models," which (...)
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  4.  14
    A Valuation Theoretic Characterization of Recursively Saturated Real Closed Fields.Paola D’Aquino, Salma Kuhlmann & Karen Lange - 2015 - Journal of Symbolic Logic 80 (1):194-206.
  5.  7
    A Characterization of the 0 -Basis Homogeneous Bounding Degrees.Karen Lange - 2010 - Journal of Symbolic Logic 75 (3):971-995.
    We say a countable model has a 0-basis if the types realized in are uniformly computable. We say has a (d-)decidable copy if there exists a model ≅ such that the elementary diagram of is (d-)computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous model with a 0-basis but no decidable copy. We extend this result here. Let d ≤ 0' be any low₂ degree. We show that there exists a homogeneous (...)
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  6.  49
    Limit Computable Integer Parts.Paola D’Aquino, Julia Knight & Karen Lange - 2011 - Archive for Mathematical Logic 50 (7-8):681-695.
    Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every ${r \in R}$ , there exists an ${i \in I}$ so that i ≤ r < i + 1. Mourgues and Ressayre (J Symb Logic 58:641–647, 1993) showed that every real closed field has an integer part. The procedure of Mourgues and Ressayre appears to be quite complicated. We would like to know whether there is a simple procedure, (...)
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  7.  16
    The Degree Spectra of Homogeneous Models.Karen Lange - 2008 - Journal of Symbolic Logic 73 (3):1009-1028.
    Much previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model A has a d-basis if the types realized in A are all computable and the Turing degree d can list $\Delta _{0}^{0}$ -indices for all types realized in A. We say A has a d-decidable copy if there exists a model B ≅ A such that the elementary (...)
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  8.  5
    -Maximal Sets.Peter A. Cholak, Peter Gerdes & Karen Lange - 2015 - Journal of Symbolic Logic 80 (4):1182-1210.
    Soare [20] proved that the maximal sets form an orbit in${\cal E}$. We consider here${\cal D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of${\cal D}$-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the${\cal D}$-maximal sets. Although these invariants help us to better understand the${\cal D}$-maximal (...)
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  9.  6
    Classifications of Computable Structures.Karen Lange, Russell Miller & Rebecca M. Steiner - 2018 - Notre Dame Journal of Formal Logic 59 (1):35-59.
    Let K be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from K such that every structure in K is isomorphic to exactly one structure on the list. Such a list is called a computable classification of K, up to isomorphism. Using the technique of Friedberg enumeration, we show that there is a computable classification of the family of computable algebraic fields and that with a 0'-oracle, we can obtain similar (...)
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  10.  7
    ${\Cal D}$-Maximal Sets.Peter A. Cholak, Peter Gerdes & Karen Lange - 2015 - Journal of Symbolic Logic 80 (4):1182-1210.
    Soare [20] proved that the maximal sets form an orbit in${\cal E}$. We consider here${\cal D}$-maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of${\cal D}$-maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the${\cal D}$-maximal sets. Although these invariants help us to better understand the${\cal D}$-maximal (...)
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  11.  9
    Erratum To: Limit Computable Integer Parts.Paola D’Aquino, Julia Knight & Karen Lange - 2015 - Archive for Mathematical Logic 54 (3-4):487-489.
  12.  6
    Representing Scott Sets in Algebraic Settings.Alf Dolich, Julia F. Knight, Karen Lange & David Marker - 2015 - Archive for Mathematical Logic 54 (5-6):631-637.
    We prove that for every Scott set S there are S-saturated real closed fields and S-saturated models of Presburger arithmetic.
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  13.  3
    On N -Tardy Sets.Peter A. Cholak, Peter M. Gerdes & Karen Lange - 2012 - Annals of Pure and Applied Logic 163 (9):1252-1270.