Browsing by Autor "Alfred Dolich"

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    A Characterization of Strongly Dependent Ordered Abelian Groups
    (National University of Colombia, 2018) Alfred Dolich; John Goodrick
    We characterize all ordered Abelian groups whose first-order theory in the language {+, <, 0} is strongly dependent. The main result of this note was obtained independently by Halevi and Hasson [7] and Farré [5].
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    DISCRETE SETS DEFINABLE IN STRONG EXPANSIONS OF ORDERED ABELIAN GROUPS
    (Cambridge University Press, 2024) Alfred Dolich; John Goodrick
    Abstract We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with a particular emphasis on the set $D'$ comprised of differences between successive elements. In particular, if the burden of the structure is at most n , then the result of applying the operation $D \mapsto D'\ n$ times must be a finite set (Theorem 1.1). In the case when the structure is densely ordered and has burden $2$ , we show that any definable unary discrete set must be definable in some elementary extension of the structure $\langle \mathbb{R}; <, +, \mathbb{Z} \rangle $ (Theorem 1.3).