One dimensional structures in o-minimal theories
Assaf Hasson, Alf Onshuus, Ya'acov Peterzil
我们证明了Zil'ber三分原理适用于所有可定义在o-极小结构中的一维结构。特别是，我们证明了任何稳定的1维结构必然是局部模。主要工具是我们开发的曲线相交理论。
We prove the Zil'ber Trichotomy Principle for all 1-dimensional structures which are definable in o-minimal ones. In particular, we show that any stable 1-dimensional structure is necessarily locally modular. The main tool is a theory for intersection of curves which we develop.
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In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablility and show that this lies strictly between VC-minimality and dp-minimality. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.
We show that there is no theory that is minimal with respect to interpretability among recursively enumerable essentially undecidable theories.
Countable models of weakly quasi-o-minimal theories II
本文确认了Martin猜想在弱准o-最小理论的一个广泛子类中的正确性。
We describe Rudin-Keisler preorders and distribution functions of numbers of limit models for quite o-minimal Ehrenfeucht theories. Decomposition formulas for these distributions are found.
We apply a general approach for distributions of binary isolating and semi-isolating formulas to the class of strongly minimal theories.
We introduce extensions by rules of the extensional level of the Minimalist Foundation which turn out to be equivalent to constructive and classical axiomatic set theories.
We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types.
Minimal types in super-dependent theories
本文给出了一个可定义在o-极小结构中的理论解释实封闭的场的充要几何条件。证明通过分析有限秩超玫瑰相关理论中的刺极小型。我们证明了这样的理论是协调的荆棘极小型，这样的类型是不稳定的，当且仅当每个非代数扩张的。我们的结论是，一个类型是稳定的，当且仅当它允许在刺最小稳定类型中进行协调。我们还证明了非平凡的刺极小稳定类型扩展了稳定集。
There exist NIP and non-NTP$_2$ theories satisfying all the following conditions: It is not o-minimal; All models are strongly locally o-minimal; It has a model which is an expansion of the linearly ordered abelian group over the reals $(\mathbb R\;;\;0,+,<)$. We construct these examples using the notion of simple product.
We show that any formula with two free variables in a VC-minimal theory has VC-codensity at most two. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acl = dcl, the VC-codensity of a formula is at most the number of free variables.
The Schröder-Bernstein property for weakly minimal theories
对于一个可数的，弱极小的理论，我们表明，Schroeder-Bernstein属性（任何两个elementarily双嵌入模型是同构的）是等价于两个条件的轨道秩1类型和属性，该理论没有无限收集的成对双嵌入，成对非同构模型。我们的结论是，可数弱极小理论，Schroeder-Bernstein性质是绝对的ZFC之间的传递模型。
Characterizing Borel Isomorphism Among Some Weakly Minimal Trivial Theories
本文刻画了某些弱极小平凡理论间的 Borel 同构关系，涵盖有限等价关系族及交叉等价关系的驯良扩张，并证明了后者的 Borel 复杂性二分法。
Minimal bounded index subgroup for dependent theories
对于依赖理论T，在C_T中，对每个型可定义群G，指数有界的型可定义子群的交是指数有界的型可定义子群。
There are no minimal effectively inseparable theories
本文针对可解释性下是否存在极小有效不可分（EI）理论的问题，提出EI的理论版本tEI，证明可解释性下无极小tEI理论，结合已有定理推导出无极小EI理论，且无极小有穷可公理化EI理论，推进了第一不完备定理的局限研究。
Unstable structures definable in o-minimal theories
设M是一个可消去矩阵的o-极小结构，N是可定义在M中的不稳定结构。则存在X，可在N中解释，使得X具有从N导出的所有结构是o-极小的。特别地，X是线性有序的。作为证明的一部分，我们证明：定理1：如果N的M-维数为1，则任何1-N-型要么是强稳定的，要么是有限的O-极小的。定理2：如果N是N-极小的，则它是1-M-维的。
We show that for a model complete strongly minimal theory whose pregeometry is flat, the recursive spectrum (SRM($T$)) is either of the form $[0,α)$ for $α\in ω+2$ or $[0,n]\cup\{ω\}$ for $n\in ω$, or $\{ω\}$, or contained in $\{0,1,2\}$. Combined with previous results, this leaves precisely 4 sets for which it is not yet determined whether each is the spectrum of a model complete strongly minimal theory with a flat pregeometry.
We conjecture that for a strongly minimal theory T in a finite signature satisfying the Zilber Trichotomy, there are only three possibilities for the recursive spectrum of T: all countable models of T are recursively presentable; none of them are recursively presentable; or only the zero-dimensional model of T is recursively presentable. We prove this conjecture for disintegrated (formerly, trivial) theories and for modular groups. The conjecture also holds via known results for fields. The conjecture remains open for finite covers of groups and fields.
Given an o-minimal structure ${\mathcal M}$ with a group operation, we show that for a properly convex subset $U$, the theory of the expanded structure ${\mathcal M}'=({\mathcal M},U)$ has definable Skolem functions precisely when ${\mathcal M}'$ is valuational. As a corollary, we get an elementary proof that the theory of any such ${\mathcal M}'$ does not satisfy definable choice.
The domination monoid in o-minimal theories
本文研究o-minimal理论中全局不变类型模支配等价的幺半群，将其计算归约为1-类型类生成问题，并在实闭域中证明成立，进而结合已有结果计算实闭赋值域中的支配幺半群。
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…
Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T'$ be a consistent theory. Then there is a complete theory $T^*$ extending $T$ such that $T$ is an open core of $T^*$, but every model of $T^*$ interprets a model of $T'$. If $T'$ is NIP, $T^*$ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.
In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
We prove a model theoretic Baire category theorem for $\tildeτ_{low}^f$-sets in a countable simple theory in which the extension property is first-order and show some of its applications. We also prove a trichotomy for minimal types in countable nfcp theories: either every type that is internal in a minimal type is essentially-1-based by means of the forking topology or $T$ interprets an infinite definable 1-based group of finite $D$-rank or $T$ interprets a strongly-minimal formula.
Countable models of weakly quasi-o-minimal theories I
本文研究了弱准o-最小理论中平凡性的性质，证明了允许定义位移的此类理论具有2^ℵ₀个可数模型。
The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. Here we investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out. We also introduce, in the context of Ban…
Definable groups and fields in t-minimal theories
本文研究t-极小理论中可定义群和域的性质，证明可定义域为有限或'大'，并为可定义群赋予自然拓扑。
A note on uniform finiteness in weakly o-minimal theories
本文研究了有序群的弱o-极小扩张中一致有限性的等价条件，并指出域扩张中一致有限性可能不成立。
Absorbed Types and Derivations in Exponential o-Minimal Theories
本文研究有序域o-极小扩张中的$\mathcal{O}$-弱直接型等对象，刻画了超序列指数理论，验证其无Tressl符号择一性反例，建立相关相容性基础理论。
Topologically 1-based T-minimal Structures
本文在驯服拓扑理论框架下，于具独立邻域性质的t-极小理论中识别拓扑1-基性分界线，并证明非平凡拓扑1-基的高度饱和结构可定义具开子集阿贝尔群，拓展群结构定理适用范围。
Given a countable o-minimal theory T, we characterize the Borel complexity of isomorphism for countable models of T up to two model-theoretic invariants. If T admits a nonsimple type, then it is shown to be Borel complete by embedding the isomorphism problem for linear orders into the isomorphism problem for models of T. This is done by constructing models with specific linear orders in the tail of the Archimedean ladder of a suitable nonsimple type. If the theory admits no nonsimple types, then we use Mayer's characterization of isomorphism for such theories to compute invariants for countabl…
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.
The aim of this note is to determine whether certain non-o-minimal expansions of o-minimal theories which are known to be NIP, are also distal. We observe that while tame pairs of o-minimal structures and the real field with a discrete multiplicative subgroup have distal theories, dense pairs of o-minimal structures and related examples do not.
We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and $P$-minimal theories.
On uniform definability of types over finite sets
本文以不可分辨序列上的类型可定义性为模板，研究了一个称为“有限集上的类型一致可定义性”（UDTFS）的公式和理论的性质。我们探讨UDTFS，并显示它是如何与模型论中的知名属性。我们记得稳定理论和弱o-极小理论有UDTFS，并且UDTFS蕴含依赖。然后，我们证明了所有的dp-极小理论有UDTFS。
Spectral Spaces in o-minimal and other NIP theories
本文利用谱拓扑研究NIP理论，在o-极小情形建立谱空间与实谱及无穷小类型的联系，并证明可定义紧致群的闭点空间同胚于无穷小类型空间；对任意NIP结构引入诚实拓扑，使不变类型集成为正规谱空间，其收缩与Simon的$F_M$收缩一致。
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…
Constructing o-minimal structures with decidable theories using generic families of functions from quasianalytic classes
令$\RR_S$表示实序域由实值函数族$S$展开，其中$S$中的每个函数定义在紧致盒上，并且是某个准解析类的成员，该准解析类在函数合成、被变量除和隐式定义函数的提取的操作下是封闭的。它表明，如果家庭$S$是通用的（这是一个特定的技术定义的超越条件），那么$\RR_S$理论是可判定的，当且仅当$S$是可计算的$C^\infty$（这意味着所有的偏导数的功能在$S$可以有效地近似）。它还表明，在一定的拓扑意义上，许多通用的，可计算的$C^\infty$家庭$S$存在。
Complexity in Tame Quantum Theories
本文提出利用o-minimal结构量化物理理论中函数与集合的逻辑信息复杂性，并建议以sharp o-minimal结构作为度量量子理论复杂性的统一框架。
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