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  1. Generic variations of models of T.Andreas Baudisch - 2002 - Journal of Symbolic Logic 67 (3):1025-1038.
    Let T be a model-complete theory that eliminates the quantifier $\exists^\infty x$ . For T we construct a theory T+ such that any element in a model of T+ determines a model of T. We show that T+ has a model companion T1. We can iterate the construction. The produced theories are investigated.
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  • Theories without the tree property of the second kind.Artem Chernikov - 2014 - Annals of Pure and Applied Logic 165 (2):695-723.
    We initiate a systematic study of the class of theories without the tree property of the second kind — NTP2. Most importantly, we show: the burden is “sub-multiplicative” in arbitrary theories ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any theory is (...)
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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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  • Dp-Minimality: Basic Facts and Examples.Alfred Dolich, John Goodrick & David Lippel - 2011 - Notre Dame Journal of Formal Logic 52 (3):267-288.
    We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian (...)
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  • Automorphisms of Homogeneous Structures.A. Ivanov - 2005 - Notre Dame Journal of Formal Logic 46 (4):419-424.
    We give an example of a simple ω-categorical theory such that for any finite set of parameters the corresponding constant expansion does not satisfy the PAPA. We describe a wide class of homogeneous structures with generic automorphisms and show that some natural reducts of our example belong to this class.
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  • From stability to simplicity.Byunghan Kim & Anand Pillay - 1998 - Bulletin of Symbolic Logic 4 (1):17-36.
    §1. Introduction. In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the simple theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular (...)
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  • Ordinal diagrams for Π3-reflection.Toshiyasu Arai - 2000 - Journal of Symbolic Logic 65 (3):1375 - 1394.
    In this paper we introduce a recursive notation system O(Π 3 ) of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of Π 3 -reflection. We show that for each $\alpha in O(Π 3 ) a set theory KP Π 3 for Π 3 -reflection proves that the initial segment of O(Π 3 ) determined by α is a well ordering. Proof theoretic study for such theories (...)
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  • Generic Expansions of Geometric Theories.Somaye Jalili, Massoud Pourmahdian & Nazanin Roshandel Tavana - forthcoming - Journal of Symbolic Logic:1-22.
    As a continuation of ideas initiated in [19], we study bi-colored (generic) expansions of geometric theories in the style of the Fraïssé–Hrushovski construction method. Here we examine that the properties $NTP_{2}$, strongness, $NSOP_{1}$, and simplicity can be transferred to the expansions. As a consequence, while the corresponding bi-colored expansion of a red non-principal ultraproduct of p-adic fields is $NTP_{2}$, the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with $\operatorname {\mathrm {bdn}}(\text (...)
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  • On Rank Not Only in Nsop Theories.Jan Dobrowolski & Daniel Max Hoffmann - forthcoming - Journal of Symbolic Logic:1-34.
    We introduce a family of local ranks $D_Q$ depending on a finite set Q of pairs of the form $(\varphi (x,y),q(y)),$ where $\varphi (x,y)$ is a formula and $q(y)$ is a global type. We prove that in any NSOP $_1$ theory these ranks satisfy some desirable properties; in particular, $D_Q(x=x)<\omega $ for any finite tuple of variables x and any Q, if $q\supseteq p$ is a Kim-forking extension of types, then $D_Q(q) (...)
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  • Companionability characterization for the expansion of an o-minimal theory by a dense subgroup.Alexi Block Gorman - 2023 - Annals of Pure and Applied Logic 174 (10):103316.
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  • Topological properties of definable sets in ordered Abelian groups of burden 2.Alfred Dolich & John Goodrick - 2023 - Mathematical Logic Quarterly 69 (2):147-164.
    We obtain some new results on the topology of unary definable sets in expansions of densely ordered Abelian groups of burden 2. In the special case in which the structure has dp‐rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle‐third set (Theorem 2.9). If it has burden 2 and both an infinite (...)
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  • Co-theory of sorted profinite groups for PAC structures.Daniel Max Hoffmann & Junguk Lee - 2023 - Journal of Mathematical Logic 23 (3).
    We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the “Weak Independence Theorem” for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property [Formula: see text]. Fourth, we describe Kim-dividing in (...)
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  • Invariant measures in simple and in small theories.Artem Chernikov, Ehud Hrushovski, Alex Kruckman, Krzysztof Krupiński, Slavko Moconja, Anand Pillay & Nicholas Ramsey - 2023 - Journal of Mathematical Logic 23 (2).
    We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups (...)
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  • On the antichain tree property.JinHoo Ahn, Joonhee Kim & Junguk Lee - 2022 - Journal of Mathematical Logic 23 (2).
    In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and [Formula: see text]-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples (...)
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  • CP‐generic expansions of models of Peano Arithmetic.Athar Abdul-Quader & James H. Schmerl - 2022 - Mathematical Logic Quarterly 68 (2):171-177.
    We study notions of genericity in models of, inspired by lines of inquiry initiated by Chatzidakis and Pillay and continued by Dolich, Miller and Steinhorn in general model‐theoretic contexts. These papers studied the theories obtained by adding a “random” predicate to a class of structures. Chatzidakis and Pillay axiomatized the theories obtained in this way. In this article, we look at the subsets of models of which satisfy the axiomatization given by Chatzidakis and Pillay; we refer to these subsets in (...)
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  • Model Theory of Fields with Finite Group Scheme Actions.Daniel Max Hoffmann & Piotr Kowalski - 2023 - Journal of Symbolic Logic 88 (4):1443-1468.
    We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.
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  • Pseudofinite difference fields and counting dimensions.Tingxiang Zou - 2021 - Journal of Mathematical Logic 21 (1):2050022.
    We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.
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  • Pseudofinite difference fields and counting dimensions.Tingxiang Zou - 2021 - Journal of Mathematical Logic 21 (1):2050022.
    We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.
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  • Neutrally expandable models of arithmetic.Athar Abdul‐Quader & Roman Kossak - 2019 - Mathematical Logic Quarterly 65 (2):212-217.
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  • Pseudofinite difference fields.Tingxiang Zou - 2019 - Journal of Mathematical Logic 19 (2):1950011.
    We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields.
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  • Amalgamation of types in pseudo-algebraically closed fields and applications.Zoé Chatzidakis - 2019 - Journal of Mathematical Logic 19 (2):1950006.
    This paper studies unbounded pseudo-algebraically closed fields and shows an amalgamation result for types over algebraically closed sets. It discusses various applications, for instance that omega-free PAC fields have the property NSOP3. It also contains a description of imaginaries in PAC fields.
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  • Expansions of o-minimal structures by dense independent sets.Alfred Dolich, Chris Miller & Charles Steinhorn - 2016 - Annals of Pure and Applied Logic 167 (8):684-706.
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  • Generic trivializations of geometric theories.Alexander Berenstein & Evgueni Vassiliev - 2014 - Mathematical Logic Quarterly 60 (4-5):289-303.
    We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T. We show that while being a trivial geometric theory, inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, the NIP and the NTP2. In particular, we show that T is strongly minimal, supersimple of SU‐rank 1, has the NIP or the NTP2 exactly when has these properties. We show that (...)
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  • Superrosy fields and valuations.Krzysztof Krupiński - 2015 - Annals of Pure and Applied Logic 166 (3):342-357.
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  • Simple stable homogeneous expansions of Hilbert spaces.Alexander Berenstein & Steven Buechler - 2004 - Annals of Pure and Applied Logic 128 (1-3):75-101.
    We study simplicity and stability in some large strongly homogeneous expansions of Hilbert spaces. Our approach to simplicity is that of Buechler and Lessmann 69). All structures we consider are shown to have built-in canonical bases.
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  • 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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  • On lovely pairs of geometric structures.Alexander Berenstein & Evgueni Vassiliev - 2010 - Annals of Pure and Applied Logic 161 (7):866-878.
    We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We use the pairs to isolate a class of geometric structures called weakly locally modular which generalizes the class of linear structures in the settings of SU-rank one theories and o-minimal theories. For o-minimal theories, we use the Peterzil–Starchenko trichotomy theorem to characterize for a sufficiently general point, the local geometry around it in terms of the thorn U-rank of its type inside a lovely pair.
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  • Model Companions of for Stable T.John T. Baldwin & Saharon Shelah - 2001 - Notre Dame Journal of Formal Logic 42 (3):129-142.
    We introduce the notion T does not omit obstructions. If a stable theory does not admit obstructions then it does not have the finite cover property (nfcp). For any theory T, form a new theory by adding a new unary function symbol and axioms asserting it is an automorphism. The main result of the paper asserts the following: If T is a stable theory, T does not admit obstructions if and only if has a model companion. The proof involves some (...)
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  • Generic Expansions of Countable Models.Silvia Barbina & Domenico Zambella - 2012 - Notre Dame Journal of Formal Logic 53 (4):511-523.
    We compare two different notions of generic expansions of countable saturated structures. One kind of genericity is related to existential closure, and another is defined via topological properties and Baire category theory. The second type of genericity was first formulated by Truss for automorphisms. We work with a later generalization, due to Ivanov, to finite tuples of predicates and functions. Let $N$ be a countable saturated model of some complete theory $T$ , and let $(N,\sigma)$ denote an expansion of $N$ (...)
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  • Ages of Expansions of ω-Categorical Structures.A. Ivanov & K. Majcher - 2007 - Notre Dame Journal of Formal Logic 48 (3):371-380.
    The age of a structure M is the set of all isomorphism types of finite substructures of M. We study ages of generic expansions of ω-stable ω-categorical structures.
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  • A Note on Weakly O-Minimal Structures and Definable Completeness.Alfred Dolich - 2007 - Notre Dame Journal of Formal Logic 48 (2):281-292.
    We consider the extent to which certain properties of definably complete structures may persist in structures which are not definably complete, particularly in the weakly o-minimal structures.
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  • Wellfoundedness proofs by means of non-monotonic inductive definitions I: Π₂⁰-operators.Toshiyasu Arai - 2004 - Journal of Symbolic Logic 69 (3):830-850.
    In this paper, we prove the wellfoundedness of recursive notation systems for reflecting ordinals up to Π₃-reflection by relevant inductive definitions.
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  • Pac Structures as Invariants of Finite Group Actions.Daniel Max Hoffmann & Piotr Kowalski - forthcoming - Journal of Symbolic Logic:1-36.
    We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property (...)
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  • Vector spaces with a dense-codense generic submodule.Alexander Berenstein, Christian D'Elbée & Evgueni Vassiliev - 2024 - Annals of Pure and Applied Logic 175 (7):103442.
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