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  1. Tame Topology over dp-Minimal Structures.Pierre Simon & Erik Walsberg - 2019 - Notre Dame Journal of Formal Logic 60 (1):61-76.
    In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.
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  • The canonical topology on dp-minimal fields.Will Johnson - 2018 - Journal of Mathematical Logic 18 (2):1850007.
    We construct a nontrivial definable type V field topology on any dp-minimal field K that is not strongly minimal, and prove that definable subsets of Kn have small boundary. Using this topology and...
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  • Decidability and classification of the theory of integers with primes.Itay Kaplan & Saharon Shelah - 2017 - Journal of Symbolic Logic 82 (3):1041-1050.
    We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th which is known to be undecidable by the works of Jockusch, Bateman, and Woods.
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  • VC-density for trees.Anton Bobkov - 2019 - Archive for Mathematical Logic 58 (5-6):587-603.
    We show that in the theory of infinite trees the VC-function is optimal. This generalizes a result of Simon showing that trees are dp-minimal.
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  • Pathological examples of structures with o‐minimal open core.Alexi Block Gorman, Erin Caulfield & Philipp Hieronymi - 2021 - Mathematical Logic Quarterly 67 (3):382-393.
    This paper answers several open questions around structures with o‐minimal open core. We construct an expansion of an o‐minimal structure by a unary predicate such that its open core is a proper o‐minimal expansion of. We give an example of a structure that has an o‐minimal open core and the exchange property, yet defines a function whose graph is dense. Finally, we produce an example of a structure that has an o‐minimal open core and definable Skolem functions, but is not (...)
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  • Vapnik–Chervonenkis Density in Some Theories without the Independence Property, II.Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson & Sergei Starchenko - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):311-363.
    We study the Vapnik–Chervonenkis density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite $\mathrm {U}$-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.
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  • Strict independence.Itay Kaplan & Alexander Usvyatsov - 2014 - Journal of Mathematical Logic 14 (2):1450008.
    We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular, it follows from our investigation that in resilient theories strict non-forking is symmetric. Based on this study, we develop notions of weight which characterize NTP2, dependence and strong dependence. Many of our proofs rely on careful analysis of sequences that witness dividing. We prove simple characterizations of such sequences in resilient theories, as well as of Morley sequences which are (...)
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  • Theories without the tree property of the second kind.Artem Chernikov - 2014 - Annals of Pure and Applied Logic 165 (2):695-723.
    We initiate a systematic study of the class of theories without the tree property of the second kind — NTP2. Most importantly, we show: the burden is “sub-multiplicative” in arbitrary theories ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any theory is (...)
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  • Distal and non-distal NIP theories.Pierre Simon - 2013 - Annals of Pure and Applied Logic 164 (3):294-318.
    We study one way in which stable phenomena can exist in an NIP theory. We start by defining a notion of ‘pure instability’ that we call ‘distality’ in which no such phenomenon occurs. O-minimal theories and the p-adics for example are distal. Next, we try to understand what happens when distality fails. Given a type p over a sufficiently saturated model, we extract, in some sense, the stable part of p and define a notion of stable independence which is implied (...)
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  • 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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  • Hereditary G-compactness.Tomasz Rzepecki - 2021 - Archive for Mathematical Logic 60 (7):837-856.
    We introduce the notion of hereditary G-compactness. We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact. Assuming that a long-standing conjecture about unstable NIP theories holds, this implies that an NIP theory is hereditarily G-compact if and only if it is stable -categorical theories). We show that if G is definable over A in a hereditarily G-compact theory, then \. We also include a (...)
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  • Non-forking and preservation of NIP and dp-rank.Pedro Andrés Estevan & Itay Kaplan - 2021 - Annals of Pure and Applied Logic 172 (6):102946.
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  • Defining integer-valued functions in rings of continuous definable functions over a topological field.Luck Darnière & Marcus Tressl - 2020 - Journal of Mathematical Logic 20 (3):2050014.
    Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or (...)
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  • 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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  • 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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  • Does weak quasi-o-minimality behave better than weak o-minimality?Slavko Moconja & Predrag Tanović - 2021 - Archive for Mathematical Logic 61 (1):81-103.
    We present a relatively simple description of binary, definable subsets of models of weakly quasi-o-minimal theories. In particular, we closely describe definable linear orders and prove a weak version of the monotonicity theorem. We also prove that weak quasi-o-minimality of a theory with respect to one definable linear order implies weak quasi-o-minimality with respect to any other such order.
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  • On forking and definability of types in some dp-minimal theories.Pierre Simon & Sergei Starchenko - 2014 - Journal of Symbolic Logic 79 (4):1020-1024.
    We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst nonforking types.
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  • Semi-Equational Theories.Artem Chernikov & Alex Mennen - forthcoming - Journal of Symbolic Logic:1-32.
    We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.
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  • On a common generalization of Shelah's 2-rank, dp-rank, and o-minimal dimension.Vincent Guingona & Cameron Donnay Hill - 2015 - Annals of Pure and Applied Logic 166 (4):502-525.
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  • On Vapnik‐Chervonenkis density over indiscernible sequences.Vincent Guingona & Cameron Donnay Hill - 2014 - Mathematical Logic Quarterly 60 (1-2):59-65.
    In this paper, we study Vapnik‐Chervonenkis density (VC‐density) over indiscernible sequences (denoted VCind‐density). We answer an open question in [1], showing that VCind‐density is always integer valued. We also show that VCind‐density and dp‐rank coincide in the natural way.
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  • Burden in Henselian valued fields.Pierre Touchard - 2023 - Annals of Pure and Applied Logic 174 (10):103318.
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  • Topological properties of definable sets in ordered Abelian groups of burden 2.Alfred Dolich & John Goodrick - 2023 - Mathematical Logic Quarterly 69 (2):147-164.
    We obtain some new results on the topology of unary definable sets in expansions of densely ordered Abelian groups of burden 2. In the special case in which the structure has dp‐rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle‐third set (Theorem 2.9). If it has burden 2 and both an infinite (...)
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  • On VC-minimal fields and dp-smallness.Vincent Guingona - 2014 - Archive for Mathematical Logic 53 (5-6):503-517.
    In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
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  • Pseudo real closed fields, pseudo p-adically closed fields and NTP2.Samaria Montenegro - 2017 - Annals of Pure and Applied Logic 168 (1):191-232.
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  • Stationarily ordered types and the number of countable models.Slavko Moconja & Predrag Tanović - 2020 - Annals of Pure and Applied Logic 171 (3):102765.
    We introduce the notions of stationarily ordered types and theories; the latter generalizes weak o-minimality and the former is a relaxed version of weak o-minimality localized at the locus of a single type. We show that forking, as a binary relation on elements realizing stationarily ordered types, is an equivalence relation and that each stationarily ordered type in a model determines some order-type as an invariant of the model. We study weak and forking non-orthogonality of stationarily ordered types, show that (...)
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  • Dp-minimality: Invariant types and dp-rank.Pierre Simon - 2014 - Journal of Symbolic Logic 79 (4):1025-1045.
    This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the -theorem holds in dp-minimal theories of small or medium directionality.In the second part, we study dp-rank in dp-minimal theories and show that it enjoys many nice properties. It is continuous, definable in families and it can be characterised geometrically with no mention of indiscernible sequences. In particular, if the structure (...)
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  • Witnessing Dp-Rank.Itay Kaplan & Pierre Simon - 2014 - Notre Dame Journal of Formal Logic 55 (3):419-429.
    We prove that in $\operatorname {NTP}_{\operatorname {2}}$ theories the dp-rank of a type can be witnessed by indiscernible sequences of tuples satisfying that type. If the type has dp-rank infinity, then this can be witnessed by singletons.
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