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Anand Pillay [102]A. Pillay [21]Ap Pillay [1]
  1.  11
    Simple Theories.Byunghan Kim & Anand Pillay - 1996 - Annals of Pure and Applied Logic 88 (2):149-164.
  2.  24
    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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  3.  12
    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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  4.  23
    Type-Definability, Compact Lie Groups, and o-Minimality.Anand Pillay - 2004 - Journal of Mathematical Logic 4 (02):147-162.
    We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures (...)
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  5.  37
    Galois Groups of First Order Theories.E. Casanovas, D. Lascar, A. Pillay & M. Ziegler - 2001 - Journal of Mathematical Logic 1 (02):305-319.
    We study the groups Gal L and Gal KP, and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP. It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.
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  6.  5
    [Omnibus Review].Anand Pillay - 1984 - Journal of Symbolic Logic 49 (1):317-321.
  7.  15
    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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  8.  15
    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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  9. First Order Topological Structures and Theories.Anand Pillay - 1987 - Journal of Symbolic Logic 52 (3):763-778.
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  10.  7
    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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  11.  16
    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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  12. Number of Countable Models.Anand Pillay - 1978 - Journal of Symbolic Logic 43 (3):492-496.
  13.  21
    Borel Equivalence Relations and Lascar Strong Types.Krzysztof Krupiński, Anand Pillay & Sławomir Solecki - 2013 - Journal of Mathematical Logic 13 (2):1350008.
    The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L of T, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well (...)
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  14.  12
    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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  15.  19
    From Stability to Simplicity.Byunghan Kim & Anand Pillay - 1998 - Bulletin of Symbolic Logic 4 (1):17-36.
  16.  62
    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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  17. Some Remarks on Definable Equivalence Relations in o-Minimal Structures.Anand Pillay - 1986 - Journal of Symbolic Logic 51 (3):709-714.
  18.  43
    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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  19. Superstable Differential Fields.A. Pillay & Ž Sokolović - 1992 - Journal of Symbolic Logic 57 (1):97-108.
  20.  13
    A Free Pseudospace.Andreas Baudisch & Anand Pillay - 2000 - Journal of Symbolic Logic 65 (1):443-460.
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  21.  17
    Additive Reducts of Real Closed Fields.David Marker, Ya'acov Peterzil & Anand Pillay - 1992 - Journal of Symbolic Logic 57 (1):109-117.
  22. Theories with Exactly Three Countable Models and Theories with Algebraic Prime Models.Anand Pillay - 1980 - Journal of Symbolic Logic 45 (2):302-310.
  23.  2
    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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  24.  16
    A Note on Groups Definable in the P -Adic Field.Anand Pillay & Ningyuan Yao - 2019 - Archive for Mathematical Logic 58 (7-8):1029-1034.
    It is known Hrushovski and Pillay that a group G definable in the field \ of p-adic numbers is definably locally isomorphic to the group \\) of p-adic points of a algebraic group H over \. We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in \ is commutative-by-finite. This result extends to groups definable in p-adically closed fields. We prove our results in the more general context (...)
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  25.  39
    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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  26.  11
    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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  27.  16
    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.
  28.  12
    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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  29.  6
    Forking, Normalization and Canonical Bases.Anand Pillay - 1986 - Annals of Pure and Applied Logic 32 (1):61-81.
  30.  2
    Discrete o-Minimal Structures.Anand Pillay & Charles Steinhorn - 1987 - Annals of Pure and Applied Logic 34 (3):275-289.
  31. 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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  32.  18
    Wilfrid Hodges. Model Theory. Encyclopedia of Mathematics and its Applications, Vol. 42, Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1993, Xiii + 772 Pp. [REVIEW]Anand Pillay - 1995 - Journal of Symbolic Logic 60 (2):689-691.
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  33.  6
    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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  34.  9
    Generic Stability and Stability.Hans Adler, Enrique Casanovas & Anand Pillay - 2014 - Journal of Symbolic Logic 79 (1):179-185.
  35.  8
    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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  36.  24
    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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  37.  39
    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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  38.  18
    Countable Models of 1-Based Theories.Anand Pillay - 1992 - Archive for Mathematical Logic 31 (3):163-169.
  39.  21
    Closed Sets and Chain Conditions in Stable Theories.Anand Pillay & Gabriel Srour - 1984 - Journal of Symbolic Logic 49 (4):1350-1362.
  40.  7
    On Theories Having Three Countable Models.Koichiro Ikeda, Akito Tsuboi & Anand Pillay - 1998 - 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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  41.  13
    Ehud Hrushovski and Boris Zilber. Zariski Geometries, Journal of the American Mathematical Society, Vol. 9 , Pp. 1–56.A. Pillay - 1999 - Journal of Symbolic Logic 64 (2):906-908.
  42.  6
    Non-Totally Transcendental Unidimensional Theories.Anand Pillay & Philipp Rothmaler - 1990 - Archive for Mathematical Logic 30 (2):93-111.
  43.  29
    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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  44. Laforte, G., See Downey, R.T. Arai, Z. Chatzidakis & A. Pillay - 1998 - Annals of Pure and Applied Logic 95:287.
     
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  45.  13
    Dimension Theory and Homogeneity for Elementary Extensions of a Model.Anand Pillay - 1982 - Journal of Symbolic Logic 47 (1):147-160.
  46.  15
    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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  47.  19
    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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  48.  27
    PAS d'Imaginaires Dans L'Infini!Anand Pillay & Bruno Poizat - 1987 - Journal of Symbolic Logic 52 (2):400-403.
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  49.  10
    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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  50.  8
    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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