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  1. On a classification of theories without the independence property.Viktor Verbovskiy - 2013 - Mathematical Logic Quarterly 59 (1-2):119-124.
    A theory is stable up to Δ if any Δ-type over a model has a few extensions up to complete types. We prove that a theory has no the independence property iff it is stable up to some Δ, where each equation image has no the independence property.
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  • On definability of types and relative stability.Viktor Verbovskiy - 2019 - Mathematical Logic Quarterly 65 (3):332-346.
    In this paper, we consider the question of definability of types in non‐stable theories. In order to do this we introduce a notion of a relatively stable theory: a theory is stable up to Δ if any Δ‐type over a model has few extensions up to complete types. We prove that an n‐type over a model of a theory that is stable up to Δ is definable if and only if its Δ‐part is definable.
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  • Linear extension operators for continuous functions on definable sets in the p‐adic context.Athipat Thamrongthanyalak - 2017 - Mathematical Logic Quarterly 63 (1-2):104-108.
    Let E be a subset of. A linear extension operator is a linear map that sends a function on E to its extension on some superset of E. In this paper, we show that if E is a semi‐algebraic or subanalytic subset of, then there is a linear extension operator such that is semi‐algebraic (subanalytic) whenever f is semi‐algebraic (subanalytic).
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  • On non-abelian C-minimal groups.Patrick Simonetta - 2003 - Annals of Pure and Applied Logic 122 (1-3):263-287.
    We investigate the structure of C-minimal valued groups that are not abelian-by-finite. We prove among other things that they are nilpotent-by-finite.
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  • On dp-minimal ordered structures.Pierre Simon - 2011 - Journal of Symbolic Logic 76 (2):448 - 460.
    We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has non-empty interior, and any theory of pure tree is dp-minimal.
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  • Abelian C-minimal groups.Patrick Simonetta - 2001 - Annals of Pure and Applied Logic 110 (1-3):1-22.
    Macpherson and Steinhorn 165–209) introduce some variants of the notion of o-minimality. One of the most interesting is C-minimality, which provides a natural setting to study algebraically closed-valued fields and some valued groups. In this paper we go further in the study of the structure of C-minimal valued groups, giving a partial characterization in the abelian case. We obtain the following principle: for abelian valued groups G for which the valuation satisfies some kind of compatibility with the multiplication by any (...)
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  • B-minimality.Raf Cluckers & François Loeser - 2007 - Journal of Mathematical Logic 7 (2):195-227.
    We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b-minimality and is based on definable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b-minimal structures. We show that b-minimality applies to the theory of Henselian valued fields of characteristic zero, generalizing work by Denef–Pas [25, 26]. Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra conditions are (...)
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  • Weakly binary expansions of dense meet‐trees.Rosario Mennuni - 2022 - Mathematical Logic Quarterly 68 (1):32-47.
    We compute the domination monoid in the theory of dense meet‐trees. In order to show that this monoid is well‐defined, we prove weak binarity of and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory from [7]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.
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  • Espaces vectoriels C-minimaux.Fares Maalouf - 2010 - Journal of Symbolic Logic 75 (2):741-758.
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  • Construction d'un groupe dans les structures C-minimales.Fares Maalouf - 2008 - Journal of Symbolic Logic 73 (3):957-968.
    We will study some aspects of the local structure of models of certain C-minimal theories. We will prove (theorem 19) that, in a sufficiently saturated C-minimal structure in which the algebraic closure has the exchange property and which is locally modular, we can construct an infinite type-definable group around any non trivial point (a term to be defined later). On va étudier ici certains aspects de la structure locale des modèles de certaines théories C-minimales. On va prouver (théorème 19) que, (...)
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  • On the structure of groups endowed with a compatible c-relation.Gabriel Lehéricy - 2018 - Journal of Symbolic Logic 83 (3):939-966.
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  • On weakly circularly minimal groups.Beibut Sh Kulpeshov & Viktor V. Verbovskiy - 2015 - Mathematical Logic Quarterly 61 (1-2):82-90.
    Here we study properties of weakly circularly minimal cyclically ordered groups. The main result of the paper is that any weakly circularly minimal cyclically ordered group is abelian.
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  • Minimality conditions on circularly ordered structures.Beibut Sh Kulpeshov & H. Dugald Macpherson - 2005 - Mathematical Logic Quarterly 51 (4):377-399.
    We explore analogues of o-minimality and weak o-minimality for circularly ordered sets. Much of the theory goes through almost unchanged, since over a parameter the circular order yields a definable linear order. Working over ∅ there are differences. Our main result is a structure theory for ℵ0-categorical weakly circularly minimal structures. There is a 5-homogeneous example which is not 6-homogeneous, but any example which is k-homogeneous for some k ≥ 6 is k-homogeneous for all k.
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  • Locally constant functions in c-minimal structures.Pablo Cubides Kovacsics - 2015 - Journal of Symbolic Logic 80 (1):207-220.
  • A criterion for uniform finiteness in the imaginary sorts.Will Johnson - 2022 - Archive for Mathematical Logic 61 (3):583-589.
    Let T be a theory. If T eliminates \, it need not follow that \ eliminates \, as shown by the example of the p-adics. We give a criterion to determine whether \ eliminates \. Specifically, we show that \ eliminates \ if and only if \ is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of \ is unknown. As an application, we show that \ eliminates \ when (...)
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  • Strongly determined types.Alexandre A. Ivanov & Dugald Macpherson - 1999 - Annals of Pure and Applied Logic 99 (1-3):197-230.
    The notion of a strongly determined type over A extending p is introduced, where p .S. A strongly determined extension of p over A assigns, for any model M )- A, a type q S extending p such that, if realises q, then any elementary partial map M → M which fixes acleq pointwise is elementary over . This gives a crude notion of independence which arises very frequently. Examples are provided of many different kinds of theories with strongly determined (...)
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  • Unexpected imaginaries in valued fields with analytic structure.Deirdre Haskell, Ehud Hrushovski & Dugald Macpherson - 2013 - Journal of Symbolic Logic 78 (2):523-542.
    We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.
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  • Cell decompositions of C-minimal structures.Deirdre Haskell & Dugald Macpherson - 1994 - Annals of Pure and Applied Logic 66 (2):113-162.
    C-minimality is a variant of o-minimality in which structures carry, instead of a linear ordering, a ternary relation interpretable in a natural way on set of maximal chains of a tree. This notion is discussed, a cell-decomposition theorem for C-minimal structures is proved, and a notion of dimension is introduced. It is shown that C-minimal fields are precisely valued algebraically closed fields. It is also shown that, if certain specific ‘bad’ functions are not definable, then algebraic closure has the exchange (...)
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  • A note on valuation definable expansions of fields.Deirdre Haskell & Dugald Macpherson - 1998 - Journal of Symbolic Logic 63 (2):739-743.
  • 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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  • Classification of ℵ0-categorical C-minimal pure C-sets.Françoise Delon & Marie-Hélène Mourgues - 2024 - Annals of Pure and Applied Logic 175 (2):103375.
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  • Abelian C-minimal valued groups.F. Delon & P. Simonetta - 2017 - Annals of Pure and Applied Logic 168 (9):1729-1782.
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  • Definable types in algebraically closed valued fields.Pablo Cubides Kovacsics & Françoise Delon - 2016 - Mathematical Logic Quarterly 62 (1-2):35-45.
    In, Marker and Steinhorn characterized models of an o‐minimal theory such that all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. In o‐minimal theories, a pair of models for which all 1‐types over M realized in N are definable has already the desired property. Although it is true that if M is an algebraically closed valued field such that all 1‐types over M are definable then (...)
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  • Presburger sets and p-minimal fields.Raf Cluckers - 2003 - Journal of Symbolic Logic 68 (1):153-162.
    We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger (...)
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  • A version of p-adic minimality.Raf Cluckers & Eva Leenknegt - 2012 - Journal of Symbolic Logic 77 (2):621-630.
    We introduce a very weak language L M on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language L M are trivial functions. We also give a definitional expansion $L\begin{array}{*{20}{c}} ' \\ M \\ \end{array} $ of L M in which K has quantifier elimination, (...)
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  • The reducts of the homogeneous binary branching c-relation.Manuel Bodirsky, Peter Jonsson & Trung van Pham - 2016 - Journal of Symbolic Logic 81 (4):1255-1297.
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  • Reducts of structures and maximal-closed permutation groups.Manuel Bodirsky & Dugald Macpherson - 2016 - Journal of Symbolic Logic 81 (3):1087-1114.
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  • An introduction to theories without the independence property.Hans Adler - unknown
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