A Note on Generic Types

Imaginaries and definable types in algebraically closed valued fields

The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from \cite{hhmcrelle}, \cite{hhm} and \cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {\em generic reparametrization}. I also try to bring out the relation to the geometry of \cite{…

Topology of definable abelian groups in o-minimal structures

In this note we show that every definably connected, definably compact abelian definable group in an o-minimal expansion of a real closed field of dimension not 4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.

关于C((t))模型论的短注

A short note on model theory of C((t))

本文在赋值环语言中研究C((t))，证明其可定义子集的可定义紧性等价于闭且无界，并评论了其上的可定义群。

一些无法合并的可定义类型

Some definable types that cannot be amalgamated

本文构造了一个理论，证明其中的可定义类型不具备合并性质，从而回答了模型论中关于类型合并的一个问题。

关于ACVF中μ-稳定子的一个注记

A note on $μ$-stabilizers in ACVF

本文研究ACVF中可定义群的μ-稳定子，证明标准无界类型对应的稳定子为无限无界可定义子群，并在线性代数群情形下证明其为可解代数子群且维数等于原类型的维数。

Chain conditions in dependent groups

In this note we prove and disprove some chain conditions in type definable and definable groups in dependent, strongly dependent and strongly^{2} dependent theories.

关于 $p$-进闭域中可定义类型的研究

Around definable types in $p$-adically closed fields

本文研究$p$-进闭域中可定义类型的性质，证明可定义类型的码可取实元组、可由值群链生成、全局类型空间严格pro-可解释、类型可沿满射提升，并证明可定义群作用的商空间可定义，且可解释拓扑空间满足第一可数性与曲线选择，统一了可定义紧致性概念。

Definable and invariant types in enrichments of NIP theories

Let T be an NIP L-theory and T' be an enrichment. We give a sufficient condition on T' for the underlying L-type of any definable (respectively invariant) type over a model of T' to be definable (respectively invariant) as an L-type. Besides, we generalise work of Simon and Starchenko on the density of definable types among non forking types to this relative setting. These results are then applied to Scanlon's model completion of valued differential fields.

Externally definable sets and dependent pairs II

We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of non-forking instances of a formula (with parameters ranging over a type-definable set) can be covered with finitely many invariant types; we give some criteria for the boundedness of an expansion by a new predicate in a distal theory; naming an arbitrary small indiscernible sequence preserves NIP, while naming a large one doesn't; there are models of NIP theorie…

Type-definable NIP fields are Artin-Schreier closed

Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p > 0$, then $K$ is Artin-Schreier closed (it has no Artin-Schreier extensions). As a consequence, $p$ does not divide the degree of any finite separable extension of $K$. This generalizes a theorem of Kaplan, Scanlon, and Wagner.

Strong density of definable types and closed ordered differential fields

The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable subset X of M^n, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

Pro-definability of spaces of definable types

We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued fields and closed ordered differential fields. Furthermore, we prove pro-definability of other distinguished subspaces, some of which have an interesting geometric interpretation. Our general strategy consists in showing that definable types are uniformly definable, a property which implies pro-definability using an argument due to E. Hrushovski and F. Loese…

On maximal stable quotients of definable groups in NIP theories

For G a group definable in an NIP theory we prove that there is a smallest type-definable subgroup H of G such that the quotient G/H is stable. This generalizes the existence of G^00, the smallest type-definable subgroup of G of bounded index.

On minimal flows and definable amenability in some distal NIP theories

We study the definable topological dynamics $(G(M), S_G(M))$ of a definable group acting on its type space, where $M$ is either an $o$-minimal structure or a $p$-adically closed field, and $G$ a definable amenable group. We focus on the problem raised by Neweslki of whether weakly generic types coincide with almost periodic types, showing that the answer is positive when $G$ has boundedly many global weakly generic types. We also give two "minimal counterexamples" where $G$ has unboundedly many global weakly generic types, extending the main results of "On minimal flows, definably amenable gro…

Definable quotients of locally definable groups

We study locally definable abelian groups $\CU$ in various settings and examine conditions under which the quotient of $\CU$ by a discrete subgroup might be definable. This turns out to be related to the existence of the type-definable subgroup $\CU^{00}$ and to the divisibility of $\CU$.

Groups definable in o-minimal structures: structure theorem, G^000, definable amenability and bounded orbits

We settle some open problems in the special case of groups in o-minimal structures, such as the equality of G^00 and G^000 and the equivalence of definable amenability and existence of a type with bounded orbit. We prove almost exactness of the G to G^00 functor. We ask further questions about types with bounded orbits in NIP theories.

On the homotopy type of definable groups in an o-minimal structure

Given a definably compact group G in a saturated o-minimal structure, there is a canonical homomorphism from G to a compact real Lie group F(G). We establish a similar result for the (o-mininimal) universal cover of a definably compact group. We also show that F(G) determines the definable homotopy type of G. A crucial step is to show that the fundamental group of an open subset of F(G) is isomorphic to the definable fundamental group of its preimage in G. Our results depend on the study of the o-minimal fundamental groupoid of G.

A note on standard systems and ultrafilters

Let $(M,\scott X) \models \ACA$ be such that $P_\scott X$, the collection of all unbounded sets in $\scott X$, admits a definable complete ultrafilter and let $T$ be a theory extending first order arithmetic coded in $\scott X$ such that $M$ thinks $T$ is consistent. We prove that there is an end-extension $N \models T$ of $M$ such that the subsets of $M$ coded in $N$ are precisely those in $\scott X$. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory $T$ extending first order arithmetic is the standard system of a recursively saturated model o…

nested PLS

In this note we will introduce a class of search problems, called nested Polynomial Local Search (nPLS) problems, and show that definable NP search problems, i.e., $Σ^b_1$-definable functions in $T^2_2$ are characterized in terms of the nested PLS.

On groups with definable $f$-generics definable in $p$-adically closed fields

The aim of this paper is to develop the theory of groups definable in the $p$-adic field ${\mathbb Q}_p$, with ``definable $f$-generics" in the sense of an ambient saturated elementary extension of ${\mathbb Q}_p$. We call such groups definable $f$-generic groups. So, by a ``definable f-generic'' or dfg group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${\mathbb Q}_p$, and the small model will be ${\mathbb Q}_p$ itself. The notion of a dfg group is dual, or rather opposite …

Volumes of definable sets in o-minimal expansions and affine GAGA theorems

In this mostly expository note, I give a very quick proof of the definable Chow theorem of Peterzil and Starchenko using the Bishop-Stoll theorem and a volume estimate for definable sets due to Nguyen and Valette. The volume estimate says that any $d$-dimensional definable subset of $S\subseteq\mathbb{R}^n$ in an o-minimal expansion of the ordered field of real numbers satisfies the inequality $\mathcal{H}^d(\{x\in S:\lVert x\rVert<r\})\leq Cr^d$, where $\mathcal{H}^d$ denotes the $d$-dimensional Hausdorff measure on $\mathbb{R}^n$ and $C$ is a constant depending on $S$. A closely related volu…

Functions continuous on curves in o-minimal structures

We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal field such that, for any bounded definable function, the germ of the function on an initial segment of the curve can be continuously extended to a closed definable set. This situation is translated into a question about types: What are the conditions on an $n$-type such that, for any bounded definable function, there is a definable set containing the type on which the function is continuous, and can be extended continuously to the set's closure? All such types are definable, and we give the precise condition…

多项式有界o-极小结构中可积轨迹的可定义性

Definability of the Integrability Locus in Polynomially Bounded o-Minimal Structures

本文整合现有文献论证，证明在实数域的任何o-极小扩张中，可定义集族对应有限体积纤维的参数集是可定义的。

On minimal flows of commutative $p$-adic groups

We study the definable topological dynamics $(G,S_G(M))$ of a definable group acting on its type space, where $M$ is a structure and $G$ is a group definable in $M$. In \cite{Newelski-I}, Newelski raised a question of whether weakly generic types coincide with almost periodic types in definable topological dynamics. In \cite{YZ-Sta}, we introduced the notion of stationarity, showing the answer is positive when $G$ is a stationary definably amenable group definable over the field of $p$-adic numbers or an $o$-minimal expansion of real closed field. In this paper, we continue with the work of \c…

A note on fsg groups in p-adically closed fields

Let $G$ be a definable group in a $p$-adically closed field $M$. We show that $G$ has finitely satisfiable generics (fsg) if and only if $G$ is definably compact. The case $M = \mathbb{Q}_p$ was previously proved by Onshuus and Pillay.

可定义（最大）余有限群的同构类型

Isomorphism types of definable (maximal) cofinitary groups

本文研究了可定义最大余有限群的同构类型，证明了特定余有限表示的存在性，并构造了相关算术最大余有限群。

Definable functions in tame expansions of algebraically closed valued fields

In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: of type (I), which hold at a neighborhood of infinity, and of type (II), which hold locally for all but finitely many points in the domain of the function. In the first part of the article, we show type (I) and (II) results concerning factorizations of definable functions over the value group. As an application, we show that tame expansions of algebraically closed valued fields having value group $\mathbb{Q}$ (like $\mathbb{C}_p$ an…

Minimal bounded index subgroup for dependent theories

For a dependent theory T, in C_T for every type definable group G, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.

关于NIP群中的f-泛型类型

On f-generic types in NIP groups

该研究刻画了NIP理论中可定义群的可定义可和性，证明其等价于存在全局f-泛型类型，还推广了相关定理并应用于dp-极小群，同时给出类型可定义群对应结果的正确证明。

Definable groups in topological fields with a generic derivation

We continue the study of a class of topological $\mathcal{L}$-fields endowed with a generic derivation $δ$, focussing on describing definable groups. We show that one can associate to an $\mathcal{L}_δ$ definable group a type $\mathcal{L}$-definable topological group. We use the group configuration tool in o-minimal structures as developed by K. Peterzil.

论 p-进闭域中可定义 f-泛型群与极小流

On definable $f$-generic groups and minimal flows in $p$-adically closed fields

本文研究 p-进闭域上可定义群的 f-泛型类型，刻画了弱泛型类型无界性与强 f-泛型存在性的等价条件，并证明了弱泛型类型的 μ-不变性及其与几乎周期类型重合的判定准则。

On stable quotients

We solve two problems from the paper "On maximal stable quotients of definable groups in NIP theories" by M. Haskel and A. Pillay, which concern maximal stable quotients of groups type-definable in NIP theories. The first result says that if $G$ is a type-definable group in a distal theory, then $G^{st}=G^{00}$ (where $G^{st}$ is the smallest type-definable subgroup with $G/G^{st}$ stable, and $G^{00}$ is the smallest type-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from $T$ to the hyperimaginary expansion $T^{heq}$. The second r…

Definable convolution and idempotent Keisler measures

We initiate a systematic study of the convolution operation on Keisler measures, generalizing the work of Newelski in the case of types. Adapting results of Glicksberg, we show that the supports of generically stable (or just definable, assuming NIP) measures are nice semigroups, and classify idempotent measures in stable groups as invariant measures on type-definable subgroups. We establish left-continuity of the convolution map in NIP theories, and use it to show that the convolution semigroup on finitely satisfiable measures is isomorphic to a particular Ellis semigroup in this context.

关于o-极小热带化的注记

A note on o-minimal tropicalizations

本文在实闭域的多项式有界o-极小扩张中，给出了可定义集热带化的多种等价描述，并推广到精细热带化情形。

Definability of groups in $\aleph_0$-stable metric structures

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \end{enumerate}

Types, transversals and definable compactness in o-minimal structures

Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in o-minimal expansions of ordered groups. Along the way we prove a parameter version for o-minimal theories of the connection between dividing and definable types known in the more general dp-minimal context [SS14], through an elementary proof that avoids the use of existing forking and VC literature. In particular we show that, if an $A$-definable family of se…

Definable types in algebraically closed valued fields

Marker and Steinhorn shown that given two models $M\prec N$ of an o-minimal theory, if all 1-types over $M$ realized in $N$ are definable, then all types over $M$ realized in $N$ are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. Although it is true that if $M$ is an algebraically closed valued field such that all 1-types over $M$ are definable then all types over $M$ definable, we build a counterexample for the relative statement, \textit{i.e.}, we show for any $n\geq 1$ that there is a pair $M\prec N$ of algebraically clos…

Definable groups for dependent and 2-dependent theories

Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\barκ$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $<\barκ$ type definable subsets there is a minimal one, i.e. their intersection has bounded index. In fact, the bound is $\leq 2^{|T|}$. We then deal with 2-dependent theories, a wider class of first order theories.

在 $p$-adic 封闭域中可定义的群

On groups definable in $p$-adically closed fields

本文研究在 $p$-adic 封闭域中可定义的群的 $dfg$/$fsg$ 分解，并证明了相关结论及 Kneser-Tits 猜想的成立。

Topological groups, μ-types and their stabilizers

We consider an arbitrary topological group $G$ definable in a structure $\mathcal M$, such that some basis for the topology of $G$ consists of sets definable in $\mathcal M$. To each such group $G$ we associate a compact $G$-space of partial types $S^μ_G(M)=\{p_μ:p\in S_G(M)\}$ which is the quotient of the usual type space $S_G(M)$ by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if $p$ is a definable type then it has a corresponding definable subgroup $Stab_μ(p)$, which is the stabilizer of $p_μ$. This group is nontrivial when $p$ is unbounde…

Abelian groups definable in $p$-adically closed fields

Recall that a group $G$ has finitely satisfiable generics ($fsg$) or definable $f$-generics ($dfg$) if there is a global type $p$ on $G$ and a small model $M_0$ such that every left translate of $p$ is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a $p$-adically closed field is an extension of a definably compact $fsg$ definable group by a $dfg$ definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where $G$ is an abelian group definable in the standard model…

Definable topological dynamics

For a group $G$ definable in a first order structure $M$ we develop basic topological dynamics in the category of definable $G$-flows. In particular, we give a description of the universal definable $G$-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of $G$, that is to the quotient $G^*/{G^*}^{00}_M$ (where $G^*$ is the interpretation of $G$ in a monster model). More generally, we obtain these results locally, i.e. in the category of $Δ$-definable $G$-flows for any fixed set $Δ$ of formulas of an…

Polynomial approximation of Berkovich spaces and definable types

We investigate affine Berkovich spaces over maximally complete fields and prove that they may be approximated by simpler spaces when the only functions we need to evaluate are polynomials of bounded degree. We derive applications to semi-algebraic sets and recover a result of E. Hrushovski and F. Loeser which claims that points of Berkovich spaces give rise to definable types (a model-theoretic notion of tameness).

在NSOP₁中寻找稳定子群

Looking for stabilizers in NSOP$\_1$

该研究针对NSOP₁理论中的可定义与型可定义群，定义Kim-稳定子群，结合Hrushovsky稳定子群证明满足假设的NSOP₁域上可定义群的有界指型可定义子群可有限对一嵌入代数群，且ω-自由PAC域的可定义群满足该条件。

On properties of (weakly) small groups

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central produ…

On the (non) superstable part of the free group

In this short note we prove that a definable set $X$ over $\mathbb F_n$ is superstable only if $X(\mathbb F_n)=X(\mathbb F_ω)$.

A note on generically stable measures and fsg groups

We prove that if μis a generically stable stable measure in a first order theory with NIP and mu(ϕ(x,b)) = 0 for all b, then μ^{(n)}(\exists y(ϕ(x_1,y)\wedge ... \wedge ϕ(x_n,y))) = 0. We deduce that if G is an fsg grooup then a definable subset X of G is generic just if every translate of X does not fork over \emptyset.

外部可定义Ramsey性质与类型空间上的不动点

The externally definable Ramsey property and fixed points on type spaces

本文研究超齐次结构的外部可定义Ramsey性质，证明其等价于类型空间子流形的不动点性质，并给出字典序乘积等例子。

关于 ZFA 中 Stone-Čech 紧致化的注记

A note on Stone-Čech compactification in ZFA

本文在 ZFA 和 $\omega$-稳定结构下，证明可定义集的 Stone-Čech 紧致化可将无限构造编码为有限构造，且该紧致化保持可定义性，进而揭示了寄存器机特定形式化的等价性。

Bounded Invariant Equivalence Relations

We study strong types and Galois groups in model theory from a topological and descriptive-set-theoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems related to cardinality and Borel cardinality of strong types, quotients of definable groups and related objecets, generalising (and often improving) essentially all hitherto known results in this area. In particular, we show that under reasonable assumptions, strong type spaces are "locally" quotients of compact Polish groups. It follows that they …

Unimodularity unified

Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied. In particular, it is shown that all notions coincide for non-multidimensional theories where the dimensions are associated to strongly minimal types.

An o-minimalist view of the group configuration

The group configuration in o-minimal structures gives rise, just like in the stable case, to a transitive action of a type-definable group on a partial type. Because $acl=dcl$ the o-minimal proof is significantly simpler than Hrushovski's original argument. Several equivalent versions, which are more suitable to the o-minimal setting, are formulated, in functional language and also in terms of a certain $4$-ary relation. In addition, the following question is considered: Can every definably connected type-definable group be definably embedded into a definable group of the same dimension? Two s…

NIP理论中不变类型的极大稳定商

Maximal stable quotients of invariant types in NIP theories

本文证明了NIP理论中不变类型的实现集上存在最精细的相对类型可定义等价关系且具有稳定商，该结论是已有关于类型可定义群极大稳定商结果的对应物。

Dp-minimality: invariant types and dp-rank

This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium directionality. In the second part, we study dp-rank in dp-minimal theories and show that it enjoys many nice properties. It is continuous, definable in families and it can be characterised geometrically with no mention of indiscernible sequences. In particular, if the structure expands a divisible ordered abelian group, then dp-rank coincides with the dimen…