18 found
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Pierre Simon [17]Pierre Henri Simon [4]Pierre-Henri Simon [2]
  1.  86
    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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  2.  18
    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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  3.  8
    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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  4.  6
    Dp-Minimal Valued Fields.Franziska Jahnke, Pierre Simon & Erik Walsberg - 2017 - Journal of Symbolic Logic 82 (1):151-165.
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  5.  9
    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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  6.  2
    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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  7.  13
    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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  8.  24
    Invariant Types in NIP Theories.Pierre Simon - 2015 - Journal of Mathematical Logic 15 (2):1550006.
    We study invariant types in NIP theories. Amongst other things: we prove a definable version of the [Formula: see text]-theorem in theories of small or medium directionality; we construct a canonical retraction from the space of [Formula: see text]-invariant types to that of [Formula: see text]-finitely satisfiable types; we show some amalgamation results for invariant types and list a number of open questions.
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  9.  5
    Exact Saturation in Simple and NIP Theories.Itay Kaplan, Saharon Shelah & Pierre Simon - 2017 - Journal of Mathematical Logic 17 (1):1750001.
    A theory [Formula: see text] is said to have exact saturation at a singular cardinal [Formula: see text] if it has a [Formula: see text]-saturated model which is not [Formula: see text]-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.
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  10.  48
    A Note on Generically Stable Measures and Fsg Groups.Ehud Hrushovski, Anand Pillay & Pierre Simon - 2012 - Notre Dame Journal of Formal Logic 53 (4):599-605.
    We prove (Proposition 2.1) that if $\mu$ is a generically stable measure in an NIP (no independence property) theory, and $\mu(\phi(x,b))=0$ for all $b$ , then for some $n$ , $\mu^{(n)}(\exists y(\phi(x_{1},y)\wedge \cdots \wedge\phi(x_{n},y)))=0$ . As a consequence we show (Proposition 3.2) that if $G$ is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and $X$ is a definable subset of $G$ , then $X$ is generic if and only if every translate of $X$ does not (...)
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  11.  14
    Adding Linear Orders.Saharon Shelah & Pierre Simon - 2012 - Journal of Symbolic Logic 77 (2):717-725.
    We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)= A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets pseudofinite arithmetic.
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  12.  4
    Definable and Invariant Types in Enrichments of Nip Theories.Silvain Rideau & Pierre Simon - 2017 - Journal of Symbolic Logic 82 (1):317-324.
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  13.  16
    Finding Generically Stable Measures.Pierre Simon - 2012 - Journal of Symbolic Logic 77 (1):263-278.
    This work builds on previous papers by Hrushovski, Pillay and the author where Keisler measures over NIP theories are studied. We discuss two constructions for obtaining generically stable measures in this context. First, we show how to symmetrize an arbitrary invariant measure to obtain a generically stable one from it. Next, we show that suitable sigma-additive probability measures give rise to generically stable Keisler measures. Also included is a proof that generically stable measures over o-minimal theories and the p-adics are (...)
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  14.  4
    Henselian Valued Fields and Inp-Minimality.Artem Chernikov & Pierre Simon - 2019 - Journal of Symbolic Logic 84 (4):1510-1526.
    We prove that every ultraproduct of p-adics is inp-minimal. More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.
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  15. Formalisme Et Signification (*).André Guimbretiere, Henri Ronse, Claude Mauriac, Maurice Mouillaud, Jean Rousset & Pierre-Henri Simon - 1965 - Cahiers Internationaux de Symbolisme 7:1.
  16.  17
    NIP Henselian Valued Fields.Franziska Jahnke & Pierre Simon - 2020 - Archive for Mathematical Logic 59 (1-2):167-178.
    We show that any theory of tame henselian valued fields is NIP if and only if the theory of its residue field and the theory of its value group are NIP. Moreover, we show that if is a henselian valued field of residue characteristic \=p\) such that if \, depending on the characteristic of K either the degree of imperfection or the index of the pth powers is finite, then is NIP iff Kv is NIP and v is roughly separably (...)
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  17.  16
    Boundedness and Absoluteness of Some Dynamical Invariants in Model Theory.Krzysztof Krupiński, Ludomir Newelski & Pierre Simon - 2019 - Journal of Mathematical Logic 19 (2):1950012.
    Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text]. By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see (...)
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  18.  1
    On Amalgamation in NTP $_{2}$ Theories and Generically Simple Generics.Pierre Simon - 2020 - Notre Dame Journal of Formal Logic 61 (2):233-243.
    We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.
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