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  1. Hyperimaginaries and Automorphism Groups.D. Lascar & A. Pillay - 2001 - Journal of Symbolic Logic 66 (1):127-143.
  • 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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  • Properties of forking in {$ømega$}-free pseudo-algebraically closed fields.Zoé Chatzidakis - 2002 - Journal of Symbolic Logic 67 (3):957-996.
  • Lovely pairs of models: the non first order case.Itay Ben-Yaacov - 2004 - Journal of Symbolic Logic 69 (3):641-662.
    We prove that for every simple theory T there is a compact abstract theory T.
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  • Neostability-properties of Fraïssé limits of 2-nilpotent groups of exponent $${p > 2}$$ p > 2.Andreas Baudisch - 2016 - Archive for Mathematical Logic 55 (3-4):397-403.
    Let L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}$$\end{document} be the language of group theory with n additional new constant symbols c1,…,cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c_1,\ldots,c_n}$$\end{document}. In L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}$$\end{document} we consider the class K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{K}}}$$\end{document} of all finite groups G of exponent p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p > 2}$$\end{document}, where G′⊆⟨c1G,…,cnG⟩⊆Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...)
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  • Product of invariant types modulo domination–equivalence.Rosario Mennuni - 2020 - Archive for Mathematical Logic 59 (1):1-29.
    We investigate the interaction between the product of invariant types and domination–equivalence. We present a theory where the latter is not a congruence with respect to the former, provide sufficient conditions for it to be, and study the resulting quotient when it is.
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  • Coordinatisation by Binding Groups and Unidimensionality in Simple Theories.Ziv Shami - 2004 - Journal of Symbolic Logic 69 (4):1221 - 1242.
    In a simple theory with elimination of finitary hyperimaginaries if tp(a) is real and analysable over a definable set Q, then there exists a finite sequence ( $a_{i}|i \leq n^{*}$ ) $\subseteq dcl^{eq}$ (a) with $a_{n}*$ = a such that for every $i \leq n*$ , if $p_{i} = tp(a_{i}/{a_{i}|j < i}$ ) then $Aut(p_{i}/Q)$ is type-definable with its action on $p_{i}^{c}$ . A unidimensional simple theory eliminates the quantifier $\exists^{\infty}$ and either interprets (in $C^{eq}$ ) an infinite type-definable group (...)
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  • Multidimensional exact classes, smooth approximation and bounded 4-types.Daniel Wolf - 2020 - Journal of Symbolic Logic 85 (4):1305-1341.
    In connection with the work of Anscombe, Macpherson, Steinhorn and the present author in [1] we investigate the notion of a multidimensional exact class, a special kind of multidimensional asymptotic class with measuring functions that yield the exact sizes of definable sets, not just approximations. We use results about smooth approximation [24] and Lie coordinatization [13] to prove the following result, as conjectured by Macpherson: For any countable language $\mathcal {L}$ and any positive integer d the class $\mathcal {C}$ of (...)
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  • Generic pairs of SU-rank 1 structures.Evgueni Vassiliev - 2003 - Annals of Pure and Applied Logic 120 (1-3):103-149.
    For a supersimple SU-rank 1 theory T we introduce the notion of a generic elementary pair of models of T . We show that the theory T* of all generic T-pairs is complete and supersimple. In the strongly minimal case, T* coincides with the theory of infinite dimensional pairs, which was used in 1184–1194) to study the geometric properties of T. In our SU-rank 1 setting, we use T* for the same purpose. In particular, we obtain a characterization of linearity (...)
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  • Constructing the hyperdefinable group from the group configuration.Tristram de Piro, Byunghan Kim & Jessica Millar - 2006 - Journal of Mathematical Logic 6 (2):121-139.
    Under [Formula: see text]-amalgamation, we obtain the canonical hyperdefinable group from the group configuration.
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  • 10th Asian Logic Conference: Sponsored by the Association for Symbolic Logic.Toshiyasu Arai - 2009 - Bulletin of Symbolic Logic 15 (2):246-265.
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  • On the forking topology of a reduct of a simple theory.Ziv Shami - 2020 - Archive for Mathematical Logic 59 (3-4):313-324.
    Let T be a simple L-theory and let \ be a reduct of T to a sublanguage \ of L. For variables x, we call an \-invariant set \\) in \ a universal transducer if for every formula \\in L^-\) and every a, $$\begin{aligned} \phi ^-\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma \wedge \phi ^-\ L\text{-forks } \text{ over }\ \emptyset. \end{aligned}$$We show that there is a greatest universal transducer \ and it is type-definable. In (...)
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  • On uncountable hypersimple unidimensional theories.Ziv Shami - 2014 - Archive for Mathematical Logic 53 (1-2):203-210.
    We extend the dichotomy between 1-basedness and supersimplicity proved in Shami :309–332, 2011). The generalization we get is to arbitrary language, with no restrictions on the topology [we do not demand type-definabilty of the open set in the definition of essential 1-basedness from Shami :309–332, 2011)]. We conclude that every hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the (...)
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  • Internality and interpretable automorphism groups in simple theories.Ziv Shami - 2004 - Annals of Pure and Applied Logic 129 (1-3):149-162.
    The binding group theorem for stable theories is partially extended to the simple context. Some results concerning internality are proved. We also introduce a ‘small’ normal subgroup G0+ of the automorphism group and show that if p is Q-internal then it has a finite exponent and G/G0+ is interpretable.
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  • Continuity of SU-rank in unidimensional supersimple theories.Ziv Shami - 2016 - Archive for Mathematical Logic 55 (5-6):663-675.
    In a supersimple unidimensional theory, SU-rank is continuous and D-rank is definable.
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  • A note on the non‐forking‐instances topology.Ziv Shami - 2020 - Mathematical Logic Quarterly 66 (3):336-340.
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  • Simple generic structures.Massoud Pourmahdian - 2003 - Annals of Pure and Applied Logic 121 (2-3):227-260.
    A study of smooth classes whose generic structures have simple theory is carried out in a spirit similar to Hrushovski 147; Simplicity and the Lascar group, preprint, 1997) and Baldwin–Shi 1). We attach to a smooth class K0, of finite -structures a canonical inductive theory TNat, in an extension-by-definition of the language . Here TNat and the class of existentially closed models of =T+,EX, play an important role in description of the theory of the K0,-generic. We show that if M (...)
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  • 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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  • Modular types in some supersimple theories.Ludomir Newelski - 2002 - Journal of Symbolic Logic 67 (4):1601-1615.
  • Adding Skolem functions to simple theories.Herwig Nübling - 2004 - Archive for Mathematical Logic 43 (3):359-370.
    We examine the conditions under which we can keep simplicity or categoricity after adding a Skolem function to the theory.
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  • On enveloping type-definable structures.Cédric Milliet - 2011 - Journal of Symbolic Logic 76 (3):1023 - 1034.
    We observe simple links between equivalence relations, groups, fields and groupoids (and between preorders, semi-groups, rings and categories), which are type-definable in an arbitrary structure, and apply these observations to the particular context of small and simple structures. Recall that a structure is small if it has countably many n-types with no parameters for each natural number n. We show that a θ-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, (...)
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  • Supersimplicity and quadratic extensions.A. Martin-Pizarro & F. O. Wagner - 2009 - Archive for Mathematical Logic 48 (1):55-61.
    An elliptic curve over a supersimple field with exactly one extension of degree 2 has an s-generic point.
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  • Forking and fundamental order in simple theories.Daniel Lascar & Anand Pillay - 1999 - Journal of Symbolic Logic 64 (3):1155-1158.
    We give a characterisation of forking in the context of simple theories in terms of the fundamental order.
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  • Disjoint $n$ -Amalgamation and Pseudofinite Countably Categorical Theories.Alex Kruckman - 2019 - Notre Dame Journal of Formal Logic 60 (1):139-160.
    Disjoint n-amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this article, we show that if a countably categorical theory T admits an expansion with disjoint n-amalgamation for all n, then T is pseudofinite. All theories which admit an expansion with disjoint n-amalgamation for all n are simple, but the method can be extended, using filtrations of Fraïssé classes, to show that certain nonsimple theories are pseudofinite. As (...)
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  • n-Simple theories.Alexei S. Kolesnikov - 2005 - Annals of Pure and Applied Logic 131 (1-3):227-261.
    The main topic of this paper is the investigation of generalized amalgamation properties for simple theories. That is, we are trying to answer the question of when a simple theory has the property of n-dimensional amalgamation, where two-dimensional amalgamation is the Independence Theorem for simple theories. We develop the notions of strong n-simplicity and n-simplicity for 1≤n≤ω, where both “1-simple” and “strongly 1-simple” are the same as “simple”. For strong n-simplicity, we present examples of simple unstable theories in each subclass (...)
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  • A note on stationarity of types over models in simple theories.Makoto Kobayashi & Akito Tsuboi - 2008 - Mathematical Logic Quarterly 54 (6):625-628.
    We investigate stationarity of types over models in simple theories. In particular, we show that in simple theories with finite SU-rank, any complete type over a model having Cantor-Bendixson rank is stationary.
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  • Simplicity, and stability in there.Byunghan Kim - 2001 - Journal of Symbolic Logic 66 (2):822-836.
    Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T, canonical base of an amalgamation class P is the union of names of ψ-definitions of P, ψ ranging over stationary L-formulas in P. Also, we prove that the same is true with stable formulas for an 1-based theory having (...)
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  • Notions around tree property 1.Byunghan Kim & Hyeung-Joon Kim - 2011 - Annals of Pure and Applied Logic 162 (9):698-709.
    In this paper, we study the notions related to tree property 1 , or, equivalently, SOP2. Among others, we supply a type-counting criterion for TP1 and show the equivalence of TP1 and k- TP1. Then we introduce the notions of weak k- TP1 for k≥2, and also supply type-counting criteria for those. We do not know whether weak k- TP1 implies TP1, but at least we prove that each weak k- TP1 implies SOP1. Our generalization of the tree-indiscernibility results in (...)
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  • Generalized amalgamation and n -simplicity.Byunghan Kim, Alexei S. Kolesnikov & Akito Tsuboi - 2008 - Annals of Pure and Applied Logic 155 (2):97-114.
    We study generalized amalgamation properties in simple theories. We formulate a notion of generalized amalgamation in such a way so that the properties are preserved when we pass from T to Teq or Theq; we provide several equivalent ways of formulating the notion of generalized amalgamation.We define two distinct hierarchies of simple theories characterized by their amalgamation properties; examples are given to show the difference between the hierarchies.
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  • A note on weak dividing.Byunghan Kim & Niandong Shi - 2007 - Archive for Mathematical Logic 46 (2):51-60.
    We study the notion of weak dividing introduced by S. Shelah. In particular we prove that T is stable iff weak dividing is symmetric.
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  • A note on Lascar strong types in simple theories.Byunghan Kim - 1998 - Journal of Symbolic Logic 63 (3):926-936.
    Let T be a countable, small simple theory. In this paper, we prove that for such T, the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.
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  • The kim–pillay theorem for abstract elementary categories.Mark Kamsma - 2020 - Journal of Symbolic Logic 85 (4):1717-1741.
    We introduce the framework of AECats, generalizing both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of models of a positive or continuous theory is an AECat. The Kim–Pillay theorem for first-order logic characterizes simple theories by the properties dividing independence has. We prove a version of the Kim–Pillay theorem for (...)
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  • Theories with equational forking.Markus Junker & Ingo Kraus - 2002 - Journal of Symbolic Logic 67 (1):326-340.
    We show that equational independence in the sense of Srour equals local non-forking. We then examine so-called almost equational theories where equational independence is a symmetric relation.
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  • Schlanke Körper (Slim fields).Markus Junker & Jochen Koenigsmann - 2010 - Journal of Symbolic Logic 75 (2):481-500.
    We examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example, they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.
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  • Recursive functions and existentially closed structures.Emil Jeřábek - 2019 - Journal of Mathematical Logic 20 (1):2050002.
    The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory T in which all partially recursive functions are representable, yet T does not interpret Robinson’s theory R. To this end, we borrow tools from model theory — specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of ∃∀ theories interpretable in (...)
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  • Independence, dimension and continuity in non-forking frames.Adi Jarden & Alon Sitton - 2013 - Journal of Symbolic Logic 78 (2):602-632.
    The notion $J$ is independent in $(M,M_0,N)$ was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal $\lambda$ and has a non-forking relation, satisfying the good $\lambda$-frame axioms and some additional hypotheses. Shelah uses independence to define dimension. Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved. As a corollary, we weaken the stability hypothesis and two additional (...)
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  • Strongly determined types.Alexandre A. Ivanov & Dugald Macpherson - 1999 - Annals of Pure and Applied Logic 99 (1-3):197-230.
    The notion of a strongly determined type over A extending p is introduced, where p .S. A strongly determined extension of p over A assigns, for any model M )- A, a type q S extending p such that, if realises q, then any elementary partial map M → M which fixes acleq pointwise is elementary over . This gives a crude notion of independence which arises very frequently. Examples are provided of many different kinds of theories with strongly determined (...)
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  • Uncountable Dense Categoricity in Cats.Itay Ben-Yaacov - 2005 - Journal of Symbolic Logic 70 (3):829 - 860.
    We prove that under reasonable assumptions, every cat (compact abstract theory) is metric, and develop some of the theory of metric cats. We generalise Morley's theorem: if a countable Hausdorff cat T has a unique complete model of density character Λ ≥ ω₁, then it has a unique complete model of density character Λ for every Λ ≥ ω₁.
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  • Simplicity and uncountable categoricity in excellent classes.Tapani Hyttinen & Olivier Lessmann - 2006 - Annals of Pure and Applied Logic 139 (1):110-137.
    We introduce Lascar strong types in excellent classes and prove that they coincide with the orbits of the group generated by automorphisms fixing a model. We define a new independence relation using Lascar strong types and show that it is well-behaved over models, as well as over finite sets. We then develop simplicity and show that, under simplicity, the independence relation satisfies all the properties of nonforking in a stable first order theory. Further, simplicity for an excellent class, as well (...)
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  • On Pseudo-Finite Dimensions.Ehud Hrushovski - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):463-495.
    We attempt to formulate issues around modularity and Zilber’s trichotomy in a setting that intersects additive combinatorics. In particular, we update the open problems on quasi-finite structures from [9].
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  • The Manin–Mumford conjecture and the model theory of difference fields.Ehud Hrushovski - 2001 - Annals of Pure and Applied Logic 112 (1):43-115.
    Using methods of geometric stability , we determine the structure of Abelian groups definable in ACFA, the model companion of fields with an automorphism. We also give general bounds on sets definable in ACFA. We show that these tools can be used to study torsion points on Abelian varieties; among other results, we deduce a fairly general case of a conjecture of Tate and Voloch on p-adic distances of torsion points from subvarieties.
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  • Model theoretic dynamics in Galois fashion.Daniel Max Hoffmann - 2019 - Annals of Pure and Applied Logic 170 (7):755-804.
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  • A geometric introduction to forking and thorn-forking.Hans Adler - 2009 - Journal of Mathematical Logic 9 (1):1-20.
    A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be (...)
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  • On positive local combinatorial dividing-lines in model theory.Vincent Guingona & Cameron Donnay Hill - 2019 - Archive for Mathematical Logic 58 (3-4):289-323.
    We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.
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  • Simple-like independence relations in abstract elementary classes.Rami Grossberg & Marcos Mazari-Armida - 2021 - Annals of Pure and Applied Logic 172 (7):102971.
  • Homology Groups of Types in Model Theory and the Computation of H 2.John Goodrick, Byunghan Kim & Alexei Kolesnikov - 2013 - Journal of Symbolic Logic 78 (4):1086-1114.
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  • Pseudofinite structures and simplicity.Darío García, Dugald Macpherson & Charles Steinhorn - 2015 - Journal of Mathematical Logic 15 (1):1550002.
    We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are (...)
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  • 2005 annual meeting of the association for symbolic logic.Ilijas Farah, Deirdre Haskell, Andrey Morozov, Vladimir Pestov & Jindrich Zapletal - 2006 - Bulletin of Symbolic Logic 12 (1):143.
  • ℵ0-categorical structures with a predimension.David M. Evans - 2002 - Annals of Pure and Applied Logic 116 (1-3):157-186.
    We give an axiomatic framework for the non-modular simple 0-categorical structures constructed by Hrushovski. This allows us to verify some of their properties in a uniform way, and to show that these properties are preserved by iterations of the construction.
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  • A Supersimple Nonlow Theory.Enrique Casanovas & Byunghan Kim - 1998 - Notre Dame Journal of Formal Logic 39 (4):507-518.
    This paper presents an example of a supersimple nonlow theory and characterizes its independence relation.
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