10 found
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  1.  24
    Dp-Minimality: Basic Facts and Examples.Alfred Dolich, John Goodrick & David Lippel - 2011 - Notre Dame Journal of Formal Logic 52 (3):267-288.
    We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian (...)
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  2.  20
    Type-amalgamation properties and polygroupoids in stable theories.John Goodrick, Byunghan Kim & Alexei Kolesnikov - 2015 - Journal of Mathematical Logic 15 (1):1550004.
    We show that in a stable first-order theory, the failure of higher dimensional type amalgamation can always be witnessed by algebraic structures that we call n-ary polygroupoids. This generalizes a result of Hrushovski in [16] that failures of 4-amalgamation are witnessed by definable groupoids. The n-ary polygroupoids are definable in a mild expansion of the language.
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  3.  10
    Homology Groups of Types in Model Theory and the Computation of H 2.John Goodrick, Byunghan Kim & Alexei Kolesnikov - 2013 - Journal of Symbolic Logic 78 (4):1086-1114.
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  4.  30
    A monotonicity theorem for dp-minimal densely ordered groups.John Goodrick - 2010 - Journal of Symbolic Logic 75 (1):221-238.
    Dp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ordered groups that are not weakly o-minimal. We introduce the even more general notion of inp-minimality and prove that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions (Theorem 3.2).
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  5.  13
    Homology groups of types in model theory and the computation of $H_2$.John Goodrick, Byunghan Kim & Alexei Kolesnikov - 2013 - Journal of Symbolic Logic 78 (4):1086-1114.
  6.  29
    Groupoids, covers, and 3-uniqueness in stable theories.John Goodrick & Alexei Kolesnikov - 2010 - Journal of Symbolic Logic 75 (3):905-929.
    Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to the (...)
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  7.  7
    Homology groups of types in stable theories and the Hurewicz correspondence.John Goodrick, Byunghan Kim & Alexei Kolesnikov - 2017 - Annals of Pure and Applied Logic 168 (9):1710-1728.
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  8. Parametric Presburger arithmetic: complexity of counting and quantifier elimination.Tristram Bogart, John Goodrick, Danny Nguyen & Kevin Woods - 2019 - Mathematical Logic Quarterly 65 (2):237-250.
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  9.  10
    Bounding quantification in parametric expansions of Presburger arithmetic.John Goodrick - 2018 - Archive for Mathematical Logic 57 (5-6):577-591.
    Generalizing Cooper’s method of quantifier elimination for Presburger arithmetic, we give a new proof that all parametric Presburger families \ [as defined by Woods ] are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f for every polynomial \. In fact, this quantifier bounding method works more generally in expansions of Presburger arithmetic by multiplication by scalars \: \alpha \in R, t \in X\}\) where R is any ring of functions from X (...)
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  10.  8
    Some remarks on inp-minimal and finite burden groups.Jan Dobrowolski & John Goodrick - 2019 - Archive for Mathematical Logic 58 (3-4):267-274.
    We prove that any left-ordered inp-minimal group is abelian and we provide an example of a non-abelian left-ordered group of dp-rank 2. Furthermore, we establish a necessary condition for a group to have finite burden involving normalizers of definable sets, reminiscent of other chain conditions for stable groups.
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