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Anand Pillay [97]A. Pillay [22]Ap Pillay [1]
  1.  4
    Simple Theories.Byunghan Kim & Anand Pillay - 1996 - Annals of Pure and Applied Logic 88 (2):149-164.
  2. Number of Countable Models.Anand Pillay - 1978 - Journal of Symbolic Logic 43 (3):492-496.
  3. Hyperimaginaries and Automorphism Groups.D. Lascar & A. Pillay - 2001 - Journal of Symbolic Logic 66 (1):127-143.
  4.  7
    Generic Structures and Simple Theories.Z. Chatzidakis & A. Pillay - 1998 - Annals of Pure and Applied Logic 95 (1-3):71-92.
    We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field is simple. 1998 Published by Elsevier Science B.V. All rights reserved.
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  5.  4
    On Compactifications and the Topological Dynamics of Definable Groups.Jakub Gismatullin, Davide Penazzi & Anand Pillay - 2014 - Annals of Pure and Applied Logic 165 (2):552-562.
    For G a group definable in some structure M, we define notions of “definable” compactification of G and “definable” action of G on a compact space X , where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as View the MathML source and the universal definable G-ambit as the type space SG. We also point out the existence and uniqueness of “universal minimal definable G-flows”, and discuss issues of amenability (...)
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  6. Coordinatisation and Canonical Bases in Simple Theories.Bradd Hart, Byunghan Kim & Anand Pillay - 2000 - Journal of Symbolic Logic 65 (1):293-309.
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  7.  4
    Lovely Pairs of Models.Itay Ben-Yaacov, Anand Pillay & Evgueni Vassiliev - 2003 - 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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  8.  7
    Type-Definability, Compact Lie Groups, and o-Minimality.Anand Pillay - 2004 - Journal of Mathematical Logic 4 (02):147-162.
  9.  2
    Superrosy Dependent Groups Having Finitely Satisfiable Generics.Clifton Ealy, Krzysztof Krupiński & Anand Pillay - 2008 - 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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  10.  2
    Connected Components of Definable Groups, and o-Minimality II.Annalisa Conversano & Anand Pillay - 2015 - Annals of Pure and Applied Logic 166 (7-8):836-849.
  11.  6
    Borel Equivalence Relations and Lascar Strong Types.Krzysztof Krupiński, Anand Pillay & Sławomir Solecki - 2013 - Journal of Mathematical Logic 13 (2):1350008.
  12. First Order Topological Structures and Theories.Anand Pillay - 1987 - Journal of Symbolic Logic 52 (3):763-778.
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  13.  1
    Topological Dynamics and Definable Groups.Anand Pillay - 2013 - 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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  14.  22
    Galois Groups of First Order Theories.E. Casanovas, D. Lascar, A. Pillay & M. Ziegler - 2001 - Journal of Mathematical Logic 1 (02):305-319.
  15. Some Remarks on Definable Equivalence Relations in o-Minimal Structures.Anand Pillay - 1986 - Journal of Symbolic Logic 51 (3):709-714.
  16. Theories with Exactly Three Countable Models and Theories with Algebraic Prime Models.Anand Pillay - 1980 - Journal of Symbolic Logic 45 (2):302-310.
  17. Superstable Differential Fields.A. Pillay & Ž Sokolović - 1992 - Journal of Symbolic Logic 57 (1):97-108.
  18.  3
    A Descending Chain Condition for Groups Definable in o-Minimal Structures.Alessandro Berarducci, Margarita Otero, Yaa’cov Peterzil & Anand Pillay - 2005 - 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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  19.  9
    From Stability to Simplicity.Byunghan Kim & Anand Pillay - 1998 - Bulletin of Symbolic Logic 4 (1):17-36.
  20. Compact Complex Manifolds with the DOP and Other Properties.Anand Pillay & Thomas Scanlon - 2002 - 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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  21.  11
    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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  22.  13
    Additive Reducts of Real Closed Fields.David Marker, Ya'Acov Peterzil & Anand Pillay - 1992 - Journal of Symbolic Logic 57 (1):109-117.
  23.  11
    A Note on CM-Triviality and the Geometry of Forking.Anand Pillay - 2000 - Journal of Symbolic Logic 65 (1):474-480.
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  24.  3
    The Definable Multiplicity Property and Generic Automorphisms.Hirotaka Kikyo & Anand Pillay - 2000 - Annals of Pure and Applied Logic 106 (1-3):263-273.
    Let T be a strongly minimal theory with quantifier elimination. We show that the class of existentially closed models of T{“σ is an automorphism”} is an elementary class if and only if T has the definable multiplicity property, as long as T is a finite cover of a strongly minimal theory which does have the definable multiplicity property. We obtain cleaner results working with several automorphisms, and prove: the class of existentially closed models of T{“σi is an automorphism”: i=1,2} is (...)
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  25.  8
    A Free Pseudospace.Andreas Baudisch & Anand Pillay - 2000 - Journal of Symbolic Logic 65 (1):443-460.
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  26. Forking, Normalization and Canonical Bases.Anand Pillay - 1986 - Annals of Pure and Applied Logic 32 (1):61-81.
  27.  5
    Remarks on Unimodularity.Charlotte Kestner & Anand Pillay - 2011 - Journal of Symbolic Logic 76 (4):1453-1458.
    We clarify and correct some statements and results in the literature concerning unimodularity in the sense of Hrushovski [7], and measurability in the sense of Macpherson and Steinhorn [8], pointing out in particular that the two notions coincide for strongly minimal structures and that another property from [7] is strictly weaker, as well as "completing" Elwes' proof [5] that measurability implies 1-basedness for stable theories.
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  28.  16
    Corps Et Chirurgie.Anand Pillay & Bruno Poizat - 1995 - 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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  29. Groups of Dimension Two and Three Over o-Minimal Structures.A. Nesin, A. Pillay & V. Razenj - 1991 - Annals of Pure and Applied Logic 53 (3):279-296.
    Let G be a group definable in an o-minimal structure M. In this paper we show: Theorem. If G is a two-dimensional definably connected nonabelian group, then G is centerless and G is isomorphic to R+R*>0, for some real closed field R. Theorem. If G is a three-dimensional nonsolvable, centerless, definably connected group, then either G SO3 or G PSL2, for some real closed field R.
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  30.  11
    Countable Models of 1-Based Theories.Anand Pillay - 1992 - Archive for Mathematical Logic 31 (3):163-169.
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  31.  3
    Stable Theories Without Dense Forking Chains.Bernhard Herwig, James G. Loveys, Anand Pillay, Predag Tanović & O. Wagner - 1992 - Archive for Mathematical Logic 31 (5):297-303.
    We define a generalized notion of rank for stable theories without dense forking chains, and use it to derive that every type is domination-equivalent to a finite product of regular types. We apply this to show that in a small theory admitting finite coding, no realisation of a nonforking extension of some strong type can be algebraic over some realisation of a forking extension.
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  32.  1
    On Definable Galois Groups and the Strong Canonical Base Property.Daniel Palacín & Anand Pillay - forthcoming - Journal of Mathematical Logic.
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  33.  2
    Non-Totally Transcendental Unidimensional Theories.Anand Pillay & Philipp Rothmaler - 1990 - Archive for Mathematical Logic 30 (2):93-111.
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  34.  16
    A Class of ℵ0-Categorical Theories.Anand Pillay - 1981 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 27 (25-30):411-418.
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  35.  10
    The Geometry of Forking and Groups of Finite Morley Rank.Anand Pillay - 1995 - 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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  36.  6
    Definability and Definable Groups in Simple Theories.Anand Pillay - 1998 - Journal of Symbolic Logic 63 (3):788-796.
    We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.
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  37.  14
    Closed Sets and Chain Conditions in Stable Theories.Anand Pillay & Gabriel Srour - 1984 - Journal of Symbolic Logic 49 (4):1350-1362.
  38. Stable Theories, Pseudoplanes and the Number of Countable Models.Anand Pillay - 1989 - 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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  39.  7
    On Lovely Pairs and the (∃ y ∈ P ) Quantifier.Anand Pillay & Evgueni Vassiliev - 2005 - 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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  40.  6
    Superstable Theories with Few Countable Models.Lee Fong Low & Anand Pillay - 1992 - 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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  41.  20
    The Prospects for Mathematical Logic in the Twenty-First Century.Samuel R. Buss, Alexander S. Kechris, Anand Pillay & Richard A. Shore - 2001 - 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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  42.  10
    On Countable Simple Unidimensional Theories.Anand Pillay - 2003 - 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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  43.  21
    1-Based Theories — the Main Gap for a -Models.B. Hart, A. Pillay & S. Starchenko - 1995 - Archive for Mathematical Logic 34 (5):285-300.
    We prove the Main Gap for the class of a -models (sufficiently saturated models) of an arbitrary stable 1-based theory T . We (i) prove a strong structure theorem for a -models, assuming NDOP, and (ii) roughly compute the number of a -models of T in any given cardinality. The analysis uses heavily group existence theorems in 1-based theories.
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  44.  7
    Differential Galois Theory II.Anand Pillay - 1997 - 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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  45.  6
    On Lascar Rank and Morley Rank of Definable Groups in Differentially Closed Fields.Anand Pillay & Wai Yan Pong - 2002 - 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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  46.  10
    Semisimple Stable and Superstable Groups.J. T. Baldwin & A. Pillay - 1989 - Annals of Pure and Applied Logic 45 (2):105-127.
  47. Discrete o-Minimal Structures.Anand Pillay & Charles Steinhorn - 1987 - Annals of Pure and Applied Logic 34 (3):275-289.
  48.  13
    Definability of Types, and Pairs of o-Minimal Structures.Anand Pillay - 1994 - 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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  49.  5
    On the Number of Models of Uncountable Theories.Ambar Chowdhury & Anand Pillay - 1994 - 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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  50.  6
    On Fields Definable inQ P.Anand Pillay - 1989 - 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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