Results for '03C64'

20 found
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  1.  9
    Definability of Henselian Valuations by Conditions on the Value Group.Lothar Sebastian Krapp, Salma Kuhlmann & Moritz Link - 2023 - Journal of Symbolic Logic 88 (3):1064-1082.
    Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any Henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for (...)
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  2.  10
    Order Types of Models of Fragments of Peano Arithmetic.Lorenzo Galeotti & Benedikt Löwe - 2022 - Bulletin of Symbolic Logic 28 (2):182-206.
    The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction (...)
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  3.  18
    Order Types of Models of Fragments of Peano Arithmetic.Lorenzo Galeotti & Benedikt Löwe - 2022 - Bulletin of Symbolic Logic 28 (2):182-206.
    The complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction (...)
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  4.  16
    Residue Field Domination in Real Closed Valued Fields.Clifton Ealy, Deirdre Haskell & Jana Maříková - 2019 - Notre Dame Journal of Formal Logic 60 (3):333-351.
    We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded T-convex (...)
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  5.  13
    On the strong cell decomposition property for weakly o‐minimal structures.Roman Wencel - 2013 - Mathematical Logic Quarterly 59 (6):452-470.
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  6.  6
    Cofinal elementary extensions.James H. Schmerl - 2014 - Mathematical Logic Quarterly 60 (1-2):12-20.
    We investigate some properties of ordered structures that are related to their having cofinal elementary extensions. Special attention is paid to models of some very weak fragments of Peano Arithmetic.
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  7.  54
    Cell decomposition and definable functions for weak p‐adic structures.Eva Leenknegt - 2012 - Mathematical Logic Quarterly 58 (6):482-497.
    We develop a notion of cell decomposition suitable for studying weak p-adic structures definable). As an example, we consider a structure with restricted addition.
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  8.  43
    Groups, group actions and fields definable in first‐order topological structures.Roman Wencel - 2012 - Mathematical Logic Quarterly 58 (6):449-467.
    Given a group , G⊆Mm, definable in a first-order structure equation image equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V⊆G and define a new topology τ on G with which becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from Mm. Likewise we topologize transitive group actions and fields definable in equation image. These results require a series of preparatory facts concerning (...)
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  9.  7
    Model Completions for Universal Classes of Algebras: Necessary and Sufficient Conditions.George Metcalfe & Luca Reggio - 2023 - Journal of Symbolic Logic 88 (1):381-417.
    Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies (...)
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  10.  4
    A Family of dp-Minimal Expansions of (Z;+).Chieu-Minh Tran & Erik Walsberg - 2023 - Notre Dame Journal of Formal Logic 64 (2):225-238.
    We consider structures of the form (Z;+,C), where C is an additive cyclic order on (Z;+). We show that such structures are dp-minimal and in this way produce a continuum-size family of dp-minimal expansions of (Z;+) such that no two members of the family define the same subsets of Z.
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  11.  8
    Strong density of definable types and closed ordered differential fields.Quentin Brouette, Pablo Cubides Kovacsics & Françoise Point - 2019 - Journal of Symbolic Logic 84 (3):1099-1117.
    The following strong form of density of definable types is introduced for theoriesTadmitting a fibered dimension functiond: given a modelMofTand a definable setX⊆Mn, there is a definable typepinX, definable over a code forXand of the samed-dimension asX. Both o-minimal theories and the theory of closed ordered differential fields are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.
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  12.  11
    Locally o-Minimal Structures with Tame Topological Properties.Masato Fujita - 2023 - Journal of Symbolic Logic 88 (1):219-241.
    We consider locally o-minimal structures possessing tame topological properties shared by models of DCTC and uniformly locally o-minimal expansions of the second kind of densely linearly ordered abelian groups. We derive basic properties of dimension of a set definable in the structures including the addition property, which is the dimension equality for definable maps whose fibers are equi-dimensional. A decomposition theorem into quasi-special submanifolds is also demonstrated.
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  13.  6
    Orthogonal Decomposition of Definable Groups.Alessandro Berarducci, Pantelis E. Eleftheriou & Marcello Mamino - forthcoming - Journal of Symbolic Logic:1-22.
    Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal (...)
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  14.  12
    Connectedness in Structures on the Real Numbers: O-Minimality and Undecidability.Alfred Dolich, Chris Miller, Alex Savatovsky & Athipat Thamrongthanyalak - 2022 - Journal of Symbolic Logic 87 (3):1243-1259.
    We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o-minimal structures on $(\mathbb {R},<)$ have the property, as do all expansions of $(\mathbb {R},+,\cdot,\mathbb {N})$. Our main analytic-geometric result is that any such expansion of $(\mathbb {R},<,+)$ by Boolean combinations of open sets (of any arities) either is o-minimal or defines an isomorph of $(\mathbb N,+,\cdot )$. We also show that any given expansion of $(\mathbb (...)
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  15.  3
    Definable Version of Wedderburn–Artin Theorem in O-Minimal Structures.Jaruwat Rodbanjong & Athipat Thamrongthanyalak - 2023 - Notre Dame Journal of Formal Logic 64 (3):353-362.
    Here we work in an arbitrary o-minimal expansion of a divisible ordered abelian group. We say that a definable ring is definably semiprime if squares of nontrivial two-sided ideals definable in the expansion are nontrivial. We prove a definable version of Wedderburn–Artin theorem and give a characterization of definably semiprime rings.
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  16.  28
    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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  17.  37
    Binary types in ℵ0‐categorical weakly o‐minimal theories.Beibut Sh Kulpeshov - 2011 - Mathematical Logic Quarterly 57 (3):246-255.
    Orthogonality of all families of pairwise weakly orthogonal 1-types for ℵ0-categorical weakly o-minimal theories of finite convexity rank has been proved in 6. Here we prove orthogonality of all such families for binary 1-types in an arbitrary ℵ0-categorical weakly o-minimal theory and give an extended criterion for binarity of ℵ0-categorical weakly o-minimal theories . © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
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  18.  17
    Definably connected nonconnected sets.Antongiulio Fornasiero - 2012 - Mathematical Logic Quarterly 58 (1):125-126.
    We give an example of a structure equation image on the real line, and a manifold M definable in equation image, such that M is definably connected but is not connected.
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  19.  1
    O-minimal de Rham Cohomology.Rodrigo Figueiredo - 2022 - Bulletin of Symbolic Logic 28 (4):529-529.
    O-minimal geometry generalizes both semialgebraic and subanalytic geometries, and has been very successful in solving special cases of some problems in arithmetic geometry, such as André–Oort conjecture. Among the many tools developed in an o-minimal setting are cohomology theories for abstract-definable continuous manifolds such as singular cohomology, sheaf cohomology and Čech cohomology, which have been used for instance to prove Pillay’s conjecture concerning definably compact groups. In the present thesis we elaborate an o-minimal de Rham cohomology theory for abstract-definable $C^{\infty (...)
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  20.  5
    Elimination of Imaginaries in Ordered Abelian Groups with Bounded Regular Rank.Mariana Vicaría - 2023 - Journal of Symbolic Logic 88 (4):1639-1654.
    In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma /\Delta $ for each definable convex subgroup $\Delta $, and sorts for the quotient groups $\Gamma /(\Delta + \ell \Gamma )$ where $\Delta $ is a definable convex subgroup and $\ell \in \mathbb {N}_{\geq 2}$. (...)
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