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.

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.

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 $\operatorname {\mathrm {bdn}}(\text (...) {"}x=x\text {"})=(\aleph _0)_{-}$. (shrink)

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 (...) is first order. (shrink)

We generalize a well-known theorem binding the elementary equivalence relation on the level of PAC fields and the isomorphism type of their absolute Galois groups. Our results concern two cases: saturated PAC structures and nonsaturated PAC structures.

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 $N$ (...) to the signature $L_{0}$ which is a model of some universal theory $T_{0}$ . We prove that when all existentially closed models of $T_{0}$ have the same existential theory, $(N,\sigma)$ is Truss generic if and only if $(N,\sigma)$ is an e-atomic model. When $T$ is $\omega$ -categorical and $T_{0}$ has a model companion $T_{\mathrm {mc}}$ , the e-atomic models are simply the atomic models of $T_{\mathrm {mc}}$. (shrink)

An amalgamation base p in a simple theory is stably definable if its canonical base is interdefinable with the set of canonical parameters for the ϕ-definitions of p as ϕ ranges through all stable formulae. A necessary condition for stably definability is given and used to produce an example of a supersimple theory with stable forking having types that are not stably definable. This answers negatively a question posed in [8]. A criterion for and example of a stably definable amalgamation (...) base whose restriction to the canonical base is not axiomatised by stable formulae are also given. The examples involve generic relations over non CM-trivial stable theories. (shrink)

§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 (...) algebraic structures which have been studied recently turn out to be simple. The interest is basically that a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. Stable theories will be defined formally in the next section. An exhaustive study of them is carried out in [33]. Without trying to read Shelah's mind, we feel comfortable in saying that the importance of stability for Shelah lay partly in the fact that an unstable theory T has 2λ many models in any cardinal λ ≥ ω1 + |T|. (shrink)

For a theory T in L, T σ is the theory of the models of T with an automorphism σ. If T is an unstable model complete theory without the independence property, then T σ has no model companion. If T is an unstable model complete theory and T σ has the amalgamation property, then T σ has no model companion. If T is model complete and has the fcp, then T σ has no model completion.

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 (...) of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler’s construction for groups, Chatzidakis’ style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras. (shrink)

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.

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a group scheme action.

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 . For any theory T, form a new theory $T_{\rm Aut}$ 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 $T_{\rm Aut}$ has a model companion. (...) The proof involves some interesting new consequences of the nfcp. (shrink)

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 (...) closed fields from [6] are also valid for pseudo-differentially closed fields. (shrink)

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.

Let G be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in m commuting derivations equipped with a G-action by differential fie...

Given a theoryTextending that of dense linear orders without endpoints, in a language ℒ ⊇ {<}, we are interested in extensionsT′ ofTin languages extending ℒ by unary relation symbols that are each interpreted in models ofT′ as sets that are both dense and codense in the underlying sets of the models.There is a canonically “wild” example, namelyT= Th andT′ = Th. Recall thatTis o-minimal, and so every open set definable in any model ofThas only finitely many definably connected components. But (...) it is well known that 〈ℝ, <, +, · ℚ 〉 defines every real Borel set, in particular, every open subset of any finite cartesian power of ℝ and every subset of any finite cartesian power of ℚ. To put this another way, the definable open sets in models ofTare essentially as simple as possible, whileT′ has a model where the definable open sets are as complicated as possible, as is the structure induced on the new predicate.In contrast to the preceding example, if ℝalgis the set of real algebraic numbers andT′ Th, then no model ofT′ defines any open set that is not definable in the underlying model ofT. (shrink)

We consider the question of when an expansion of a first-order topological structure has the property that every open set definable in the expansion is definable in the original structure. This question has been investigated by Dolich, Miller and Steinhorn in the setting of ordered structures as part of their work on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give (...) a further characterisation in the special case of an expansion by a generic predicate. (shrink)

Let${\mathbb K}$be an algebraically bounded structure, and letTbe its theory. IfTis model complete, then the theory of${\mathbb K}$endowed with a derivation, denoted by$T^{\delta }$, has a model completion. Additionally, we prove that if the theoryTis stable/NIP then the model completion of$T^{\delta }$is also stable/NIP. Similar results hold for the theory with several derivations, either commuting or non-commuting.

We define the interpolative fusion T∪∗ of a family i∈I of first-order theories over a common reduct T∩, a notion that generalizes many examples of random or generic structures in the model-theo...

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm {ACFG}$. This theory was introduced in [16] as a new example of $\mathrm {NSOP}_{1}$ nonsimple theory. In this paper we describe more features of $\mathrm {ACFG}$, such as imaginaries. We also study various independence relations in $\mathrm {ACFG}$, such as Kim-independence or forking independence, and describe interactions between them.

We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.

We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately $aleph_0$-saturated up to arbitrarily small perturbations are isomorphic up to arbitrarily small perturbations. As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories all of whose separable models are isomorphic up to arbitrarily small perturbations.

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 (...) these PAC substructures and show several results related to the SOPnhierarchy. Fifth, we characterize the algebraic closure in PAC structures. (shrink)

Nous isolons des propriétés valables dans certaines théories de purs corps ou de corps munis d’opérateurs afin de montrer qu’une théorie est simple lorsque les clôtures définissables et algébriques sont contrôlées par une théorie stable associée.

We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: "weak one-basedness", absence of type definable "almost quasidesigns", and "generic linearity". Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over (...) a finite field. (shrink)

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 (...) group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p -adic numbers is dp-minimal. (shrink)

Dp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ordered groups that are not weakly o-minimal. We introduce the even more general notion of inp-minimality and prove that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions (Theorem 3.2).

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.

We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th which is known to be undecidable by the works of Jockusch, Bateman, and Woods.

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.

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 (...) an elementary class if and only if T has the definable multiplicity property. (shrink)

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 (...) always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic is NTP2 if and only if the residue field is NTP2 , so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2. (shrink)

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.

Based on the work done in [][] in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking‐independent elements that is dense inside a partial type, which we call H‐structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again H‐structures. We prove that under these assumptions the expansion is supersimple and (...) characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one‐basedness and CM‐triviality. In the one‐based case, when T has SU‐rank and the SU‐rank is continuous, we take to be the type of elements of SU‐rank and we describe a natural “geometry of generics modulo H” associated with such expansions and show it is modular. (shrink)

This article addresses the question which structures occur as fixed structures of stable structures with a generic automorphism. In particular we give a Galois theoretic characterization. Furthermore, we prove that any pseudofinite field is the fixed field of some model ofACFA, any one-free pseudo-differentially closed field of characteristic zero is the fixed field of some model ofDCFA, and that any one-free PAC field of finite degree of imperfection is the fixed field of some model ofSCFA.

Let TP be the theory obtained by adding a generic predicate to an o-minimal theory T. We prove that if T admits elimination of imaginaries, then TP also admits elimination of imaginaries.

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) (...) Kim-non-forking extension, then $D_Q(q)=D_Q(p)$ for every Q that involves only invariant types whose Morley powers are -stationary. We give natural examples of families of invariant types satisfying this property in some NSOP $_1$ theories. We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory $T_\infty $ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in $T_\infty $, strengthening an earlier observation that $T_\infty $ satisfies the existence axiom for forking independence. Finally, we slightly modify our definitions and go beyond NSOP $_1$ to find out that our local ranks are bounded by the well-known ranks: the inp-rank (burden), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example, if T is dp-minimal. Hence, our notion of rank identifies a non-trivial class of theories containing all NSOP $_1$ and NTP $_2$ theories. (shrink)