76 found
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  1.  41
    Anand Pillay (1978). Number of Countable Models. Journal of Symbolic Logic 43 (3):492-496.
  2.  0
    Anand Pillay (2013). Topological Dynamics and Definable Groups. Journal of Symbolic Logic 78 (2):657-666.
    We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group $G(M)$ on its “external type space” $S_{G,\textit{ext}}(M)$, can explain, account for, or give rise to, the quotient $G/G^{00}$, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or $fsg$) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of $G(M)$ (...)
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  3.  0
    Byunghan Kim & Anand Pillay (1997). Simple Theories. Annals of Pure and Applied Logic 88 (2):149-164.
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  4.  2
    Itay Ben-Yaacov, Anand Pillay & Evgueni Vassiliev (2003). Lovely Pairs of Models. Annals of Pure and Applied Logic 122 (1-3):235-261.
    We introduce the notion of a lovely pair of models of a simple theory T, generalizing Poizat's “belles paires” of models of a stable theory and the third author's “generic pairs” of models of an SU-rank 1 theory. We characterize when a saturated model of the theory TP of lovely pairs is a lovely pair , finding an analog of the nonfinite cover property for simple theories. We show that, under these hypotheses, TP is also simple, and we study forking (...)
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  5.  83
    Bradd Hart, Byunghan Kim & Anand Pillay (2000). Coordinatisation and Canonical Bases in Simple Theories. Journal of Symbolic Logic 65 (1):293-309.
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  6.  4
    Anand Pillay (1982). Dimension Theory and Homogeneity for Elementary Extensions of a Model. Journal of Symbolic Logic 47 (1):147-160.
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  7.  2
    Anand Pillay (2004). Type-Definability, Compact Lie Groups, and o-Minimality. Journal of Mathematical Logic 4 (02):147-162.
  8.  75
    Anand Pillay (1987). First Order Topological Structures and Theories. Journal of Symbolic Logic 52 (3):763-778.
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  9.  3
    Alessandro Berarducci, Margarita Otero, Yaa’cov Peterzil & Anand Pillay (2005). A Descending Chain Condition for Groups Definable in o-Minimal Structures. Annals of Pure and Applied Logic 134 (2):303-313.
    We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest type-definable subgroup G00 of bounded index and G/G00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author.
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  10.  2
    Clifton Ealy, Krzysztof Krupiński & Anand Pillay (2008). Superrosy Dependent Groups Having Finitely Satisfiable Generics. Annals of Pure and Applied Logic 151 (1):1-21.
    We develop a basic theory of rosy groups and we study groups of small Uþ-rank satisfying NIP and having finitely satisfiable generics: Uþ-rank 1 implies that the group is abelian-by-finite, Uþ-rank 2 implies that the group is solvable-by-finite, Uþ-rank 2, and not being nilpotent-by-finite implies the existence of an interpretable algebraically closed field.
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  11.  43
    Anand Pillay (1986). Some Remarks on Definable Equivalence Relations in o-Minimal Structures. Journal of Symbolic Logic 51 (3):709-714.
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  12.  41
    Anand Pillay (1980). Theories with Exactly Three Countable Models and Theories with Algebraic Prime Models. Journal of Symbolic Logic 45 (2):302-310.
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  13.  10
    David Marker, Ya'Acov Peterzil & Anand Pillay (1992). Additive Reducts of Real Closed Fields. Journal of Symbolic Logic 57 (1):109-117.
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  14.  16
    Samuel R. Buss, Alexander S. Kechris, Anand Pillay & Richard A. Shore (2001). The Prospects for Mathematical Logic in the Twenty-First Century. Bulletin of Symbolic Logic 7 (2):169-196.
    The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
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  15.  64
    Anand Pillay & Thomas Scanlon (2002). Compact Complex Manifolds with the DOP and Other Properties. Journal of Symbolic Logic 67 (2):737-743.
    We point out that a certain complex compact manifold constructed by Lieberman has the dimensional order property, and has U-rank different from Morley rank. We also give a sufficient condition for a Kahler manifold to be totally degenerate (that is, to be an indiscernible set, in its canonical language) and point out that there are K3 surfaces which satisfy these conditions.
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  16.  5
    Anand Pillay (2003). On Countable Simple Unidimensional Theories. Journal of Symbolic Logic 68 (4):1377-1384.
    We prove that any countable simple unidimensional theory T is supersimple, under the additional assumptions that T eliminates hyperimaginaries and that the $D_\phi-ranks$ are finite and definable.
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  17.  6
    Anand Pillay & Wai Yan Pong (2002). On Lascar Rank and Morley Rank of Definable Groups in Differentially Closed Fields. Journal of Symbolic Logic 67 (3):1189-1196.
    Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.
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  18.  2
    Anand Pillay (1992). Countable Models of 1-Based Theories. Archive for Mathematical Logic 31 (3):163-169.
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  19.  4
    Ambar Chowdhury & Anand Pillay (1994). On the Number of Models of Uncountable Theories. Journal of Symbolic Logic 59 (4):1285-1300.
    In this paper we establish the following theorems. THEOREM A. Let T be a complete first-order theory which is uncountable. Then: (i) I(|T|, T) ≥ ℵ 0 . (ii) If T is not unidimensional, then for any λ ≥ |T|, I (λ, T) ≥ ℵ 0 . THEOREM B. Let T be superstable, not totally transcendental and nonmultidimensional. Let θ(x) be a formula of least R ∞ rank which does not have Morley rank, and let p be any stationary completion (...)
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  20.  16
    Anand Pillay & Bruno Poizat (1995). Corps Et Chirurgie. Journal of Symbolic Logic 60 (2):528-533.
    Les corps algébriquement clos, réels clos et pseudo-finis n'ont, pour chaque entier n, qu'un nombre fini d'extensions de degré n; nous montrons qu'ils partagent cette propriété avec tous les corps qui, comme eux, satisfont une propriété très rudimentaire de préservation de la dimension, de nature modèle-théorique. Ce résultat est atteint en montrant qu'une certaine action du groupe GLn d'un tel corps n'a qu'un nombre fini d'orbites. /// La korpoj algebre fermataj, reale fermataj kaj pseudofinataj ne havas, pri ciu integro n, (...)
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  21.  0
    Anand Pillay (1989). Stable Theories, Pseudoplanes and the Number of Countable Models. Annals of Pure and Applied Logic 43 (2):147-160.
    We prove that if T is a stable theory with only a finite number of countable models, then T contains a type-definable pseudoplane. We also show that for any stable theory T either T contains a type-definable pseudoplane or T is weakly normal.
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  22.  0
    Anand Pillay (1986). Forking, Normalization and Canonical Bases. Annals of Pure and Applied Logic 32 (1):61-81.
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  23.  7
    Anand Pillay (1984). Regular Types in Nonmultidimensional Ω-Stable Theories. Journal of Symbolic Logic 49 (3):880-891.
    We define a hierarchy on the regular types of an ω-stable nonmultidimensional theory, using generalised notions of algebraic and strongly minimal formulae. As an application we show that any resplendent model of an ω-stable finite-dimensional theory is saturated.
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  24.  0
    Koichiro Ikeda, Akito Tsuboi & Anand Pillay (1998). On Theories Having Three Countable Models. Mathematical Logic Quarterly 44 (2):161-166.
    A theory T is called almost [MATHEMATICAL SCRIPT CAPITAL N]0-categorical if for any pure types p1,…,pn there are only finitely many pure types which extend p1 ∪…∪pn. It is shown that if T is an almost [MATHEMATICAL SCRIPT CAPITAL N]0-categorical theory with I = 3, then a dense linear ordering is interpretable in T.
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  25.  12
    Anand Pillay & Gabriel Srour (1984). Closed Sets and Chain Conditions in Stable Theories. Journal of Symbolic Logic 49 (4):1350-1362.
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  26.  6
    Anand Pillay (1989). On Fields Definable inQ P. Archive for Mathematical Logic 29 (1):1-7.
    We prove that any field definable in (Q p, +, ·) is definably isomorphic to a finite extension ofQ p.
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  27.  5
    Anand Pillay (1991). Some Remarks on Modular Regular Types. Journal of Symbolic Logic 56 (3):1003-1011.
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  28.  1
    Anand Pillay & Philipp Rothmaler (1990). Non-Totally Transcendental Unidimensional Theories. Archive for Mathematical Logic 30 (2):93-111.
    No categories
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  29.  1
    Anand Pillay & Akito Tsuboi (1997). Amalgamations Preserving ℵ0-Categoricity. Journal of Symbolic Logic 62 (4):1070 - 1074.
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  30.  1
    Anand Pillay (1990). Differentially Algebraic Group Chunks. Journal of Symbolic Logic 55 (3):1138-1142.
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  31.  9
    Ehud Hrushovski, Anand Pillay & Pierre Simon (2012). A Note on Generically Stable Measures and Fsg Groups. 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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  32.  4
    Anand Pillay (1988). Sheaves of Continuous Definable Functions. Journal of Symbolic Logic 53 (4):1165-1169.
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  33.  4
    Anand Pillay (2013). Weight and Measure in NIP Theories. Notre Dame Journal of Formal Logic 54 (3-4):567-578.
    We initiate an account of Shelah’s notion of “strong dependence” in terms of generically stable measures, proving a measure analogue of the fact that a stable theory $T$ is “strongly dependent” if and only if all types have almost finite weight.
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  34.  6
    Anand Pillay (1995). The Geometry of Forking and Groups of Finite Morley Rank. Journal of Symbolic Logic 60 (4):1251-1259.
    The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical group has this property. Here we show that any group of finite Morley rank definable in a CM-trivial theory is nilpotent-by-finite, or equivalently no simple group of finite Morley rank can be definable in a CM-trivial theory.
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  35.  4
    Patricia Blanchette, Kit Fine, Heike Mildenberger, André Nies, Anand Pillay, Alexander Razborov, Alexandra Shlapentokh, John R. Steel & Boris Zilber (2009). Notre Dame, Indiana May 20–May 23, 2009. Bulletin of Symbolic Logic 15 (4).
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  36.  12
    Anand Pillay & Dominika Polkowska (2006). On PAC and Bounded Substructures of a Stable Structure. Journal of Symbolic Logic 71 (2):460 - 472.
    We introduce and study the notions of a PAC-substructure of a stable structure, and a bounded substructure of an arbitrary substructure, generalizing [10]. We give precise definitions and equivalences, saying what it means for properties such as PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a "descent theorem" for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically (...)
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  37.  3
    Anand Pillay (1997). Differential Galois Theory II. Annals of Pure and Applied Logic 88 (2-3):181-191.
    First, it is pointed out how the author's new differential Galois theory contributes to the understanding of the differential closure of an arbitrary differential field . Secondly, it is shown that a superstable differential field has no proper differential Galois extensions.
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  38.  3
    Anand Pillay (1995). Review: Wilfrid Hodges, Model Theory. [REVIEW] Journal of Symbolic Logic 60 (2):689-691.
  39.  3
    Anand Pillay (1994). Some Remarks on Nonmultidimensional Superstable Theories. Journal of Symbolic Logic 59 (1):151-165.
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  40.  3
    Anand Pillay & Mark D. Schlatter (2002). Some Results on Permutation Group Isomorphism and Categoricity. Journal of Symbolic Logic 67 (3):910-914.
    We extend Morley's Theorem to show that if a theory is κ-p-categorical for some uncountable cardinal κ, it is uncountably categorical. We then discuss ω-p-categoricity and provide examples to show that similar extensions for the Baldwin-Lachlan and Lachlan Theorems are not possible.
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  41.  8
    Anand Pillay (2000). A Note on CM-Triviality and the Geometry of Forking. Journal of Symbolic Logic 65 (1):474-480.
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  42.  4
    Anand Pillay & Evgueni Vassiliev (2005). On Lovely Pairs and the (∃ y ∈ P ) Quantifier. Notre Dame Journal of Formal Logic 46 (4):491-501.
    Given a lovely pair P ≺ M of models of a simple theory T, we study the structure whose universe is P and whose relations are the traces on P of definable (in ℒ with parameters from M) sets in M. We give a necessary and sufficient condition on T (which we call weak lowness) for this structure to have quantifier-elimination. We give an example of a non-weakly-low simple theory.
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  43.  12
    Anand Pillay (1994). Definability of Types, and Pairs of o-Minimal Structures. Journal of Symbolic Logic 59 (4):1400-1409.
    Let T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L * be L together with a unary predicate P. Let T * be the L * -theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an |M| + -saturated elementary extension of N (and (...)
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  44.  6
    Anand Pillay (2001). A Note on Existentially Closed Difference Fields with Algebraically Closed Fixed Field. Journal of Symbolic Logic 66 (2):719-721.
    We point out that the theory of difference fields with algebraically closed fixed field has no model companion.
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  45.  7
    Anand Pillay & Bruno Poizat (1987). PAS d'Imaginaires Dans L'Infini! Journal of Symbolic Logic 52 (2):400-403.
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  46.  7
    Anand Pillay & Charles Steinhorn (1987). On Dedekind Complete o-Minimal Structures. Journal of Symbolic Logic 52 (1):156-164.
    For a countable complete o-minimal theory T, we introduce the notion of a sequentially complete model of T. We show that a model M of T is sequentially complete if and only if $\mathscr{M} \prec \mathscr{N}$ for some Dedekind complete model N. We also prove that if T has a Dedekind complete model of power greater than 2 ℵ 0 , then T has Dedekind complete models of arbitrarily large powers. Lastly, we show that a dyadic theory--namely, a theory relative (...)
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  47.  2
    Anand Pillay & Philipp Rothmaler (1993). Unidimensional Modules: Uniqueness of Maximal Non-Modular Submodels. Annals of Pure and Applied Logic 62 (2):175-181.
    We characterize the non-modular models of a unidimensional first-order theory of modules as the elementary submodels of its prime pure-injective model. We show that in case the maximal non-modular submodel of a given model splits off this is true for every such submodel, and we thus obtain a cancellation result for this situation. Although the theories in question always have models whose maximal non-modular submodel do split off, they may as well have others where they don't. We present a corresponding (...)
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  48.  2
    Lee Fong Low & Anand Pillay (1992). Superstable Theories with Few Countable Models. Archive for Mathematical Logic 31 (6):457-465.
    We prove here:Theorem. LetT be a countable complete superstable non ω-stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG eq orthogonal to the generics has Morley rank.Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.
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  49.  2
    S. Barry Cooper, Herman Geuvers, Anand Pillay & Jouko Väänänen (2008). Preface. Annals of Pure and Applied Logic 156 (1):1-2.
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  50.  2
    Anand Pillay (1983). A Note on Finitely Generated Models. Journal of Symbolic Logic 48 (1):163-166.
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