Classification of $\aleph_0$-categorical $C$-minimal pure $C$-sets

The complexity of the index sets of $\aleph_0$-categorical theories and of Ehrenfeucht theories

We classify the computability-theoretic complexity of two index sets of classes of first-order theories: We show that the property of being an $\aleph_0$-categorical theory is $Π^0_3$-complete; and the property of being an Ehrenfeucht theory $Π^1_1$-complete. We also show that the property of having continuum many models is $Σ^1_1$-hard. Finally, as a corollary, we note that the properties of having only decidable models, and of having only computable models, are both $Π^1_1$-complete.

Universal classes near $\aleph_1$

Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, …

Aleph-zero-categorical groups and their completions

We embed a countably categorical group G into a locally compact group c(G) with a non-trivial topology and study how topological properties of c(G) are connected with the structure of definable subgroups of G.

$\aleph_0$-categorical strongly minimal compact complex manifolds

Essential $\aleph_0$-categoricity; i.e., $\aleph_0$-categoricity in some full countable language, is shown to be a robust notion for strongly minimal compact complex manifolds. Characterisations of triviality and essential $\aleph_0$-categoricity are given in terms of complex-analytic automorphisms, in the simply connected case, and correspondences in general. As a consequence it is pointed out that an example of McMullen yields a strongly minimal compact Kähler manifold with trivial geometry but which is not $\aleph_0$-categorical, giving a counterexample to a conjecture of the second author …

$\aleph_0$-categoricity of semigroups II

A countable semigroup is $\aleph_0$-categorical if it can be characterised, up to isomorphism, by its first-order properties. In this paper we continue our investigation into the $\aleph_0$-categoricity of semigroups. Our main results are a complete classification of $\aleph_0$-categorical orthodox completely 0-simple semigroups, and descriptions of the $\aleph_0$-categorical members of certain classes of strong semilattices of semigroups.

局部 $\aleph_0$-范畴理论与局部罗尔克预紧群

Locally $\aleph_0$-categorical theories and locally Roelcke precompact groups

本文研究了局部 $\aleph_0$-范畴理论与局部罗尔克预紧群的对应关系，证明了它们的双可解释性及自同构群的同构性。

Siblings of an $\aleph_0$-categorical relational structure

A sibling of a relational structure $R$ is any structure $S$ which can be embedded into $R$ and, vice versa, in which $R$ can be embedded. Let $sib(R)$ be the number of siblings of $R$, these siblings being counted up to isomorphism. Thomassé conjectured that for countable relational structures made of at most countably many relations, $sib(R)$ is either $1$, countably infinite, or the size of the continuum; but even showing the special case $sib(R)=1$ or infinite is unsettled when $R$ is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree $T$ the number…

Approximate isomorphism of randomization pairs

We study approximate $\aleph_0$-categoricity of theories of beautiful pairs of randomizations, in the sense of continuous logic. This leads us to disprove a conjecture of Ben Yaacov, Berenstein and Henson, by exhibiting $\aleph_0$-categorical, $\aleph_0$-stable metric theories $Q$ for which the corresponding theory $Q_P$ of beautiful pairs is not approximately $\aleph_0$-categorical, i.e., has separable models that are not isomorphic even up to small perturbations of the smaller model of the pair. The theory $Q$ of randomized infinite vector spaces over a finite field is such an example. On th…

An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction

We prove an algebraic preservation theorem for positive Horn definability in aleph-zero categorical structures. In particular, we define and study a construction which we call the periodic power of a structure, and define a periomorphism of a structure to be a homomorphism from the periodic power of the structure to the structure itself. Our preservation theorem states that, over an aleph-zero categorical structure, a relation is positive Horn definable if and only if it is preserved by all periomorphisms of the structure. We give applications of this theorem, including a new proof of the know…

树的渐近类与 $\aleph_0$-范畴性

Asymptotic Classes of Trees and $\aleph_0$-categoricity

本文引入树计划刻画树的 $\aleph_0$-范畴理论，证明相关 Fraïssé 类为渐近类，并刻画有限树超积及超简单有限秩树。

$\aleph_0$-categorical Banach spaces contain $\ell_p$ or $c_0$

This paper has three parts. First, we establish some of the basic model theoretic facts about $M_{\mathcal{T}}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) $M_{\mathcal{T}}$ has the \emph{non independence property} (NIP); (2) every Banach space that is $\aleph_0$-categorical up to small perturbations embeds $c_0$ or $\ell_p$ ($1\leqslant p<\infty$) almost isometrically; consequ…

$\aleph_0$-categoricity of semigroups

In this paper we initiate the study of $\aleph_0$-categorical semigroups, where a countable semigroup $S$ is $\aleph_0$-categorical if, for any natural number $n$, the action of its group of automorphisms Aut $S$ on $S^n$ has only finitely many orbits. We show that $\aleph_0$-categoricity transfers to certain important substructures such as maximal subgroups and principal factors. We examine the relationship between $\aleph_0$-categoricity and a number of semigroup and monoid constructions, namely direct sums, 0-direct unions, semidirect products and $\mathcal{P}$-semigroups. As a corollary, w…

何时$\u200e\aleph_1$-范畴性蕴含$\omega$-稳定性？

When does $\aleph_1$-categoricity imply $ω$-stability?

本研究针对ℵ₁-范畴的原子类，厘清ℵ₁大小唯一模型上的型空间，证明若此类存在大小为ℶ₁⁺的模型则具有ω-稳定性，明确了ℵ₁-范畴性蕴含ω-稳定性的条件。

Star 分类、Lelek 扇形与连续逻辑中非$\boldsymbol{\nwarrow\nearrow}_0$-范畴理论的重构

Star sorts, Lelek fans, and the reconstruction of non-$\aleph\_0$-categorical theories in continuous logic

本文针对可数语言下连续/经典逻辑的任意理论，构建了以开波兰拓扑群胚$\boldsymbol{G}^*(T)$为形式的双解释不变量，其基与Lelek扇形同胚，通过两步构造法将结果纳入统一框架。

Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups

We investigate the automorphism groups of $\aleph\_0$-categorical structures and prove that they are exactly the Roelcke precompact Polish groups. We show that the theory of a structure is stable if and only if every Roelcke uniformly continuous function on the automorphism group is weakly almost periodic. Analysing the semigroup structure on the weakly almost periodic compactification, we show that continuous surjective homomorphisms from automorphism groups of stable $\aleph\_0$-categorical structures to Hausdorff topological groups are open. We also produce some new WAP-trivial groups and c…

Automorphism groups of randomized structures

We study automorphism groups of randomizations of separable structures, with focus on the $\aleph_0$-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the $\aleph_0$-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and cl…

Almost disjoint pure subgroups of the Baer-Specker group

We prove in ZFC that the Baer-Specker group Z^omega has 2^{aleph_1} non-free pure subgroups of cardinality aleph_1 which are almost disjoint: there is no non-free subgroup embeddable in any pair.

The Borel Complexity of Isomorphism for Complete Theories of Linear Orders With Unary Predicates

We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is $\aleph_0$-categorical, has finitely many countable models (at least three), is Borel equivalent to equality on the reals, is Borel equivalent to "countable sets of reals," or is Borel complete. Furthermore, each of these cases corresponds in a natural way to a count of models of all sizes, up to back-and-forth equivalence. All these cases are possible and we compute precise…

非初等范畴性与射影局部 o-极小类

Non-elementary categoricity and projective locally o-minimal classes

本文将光滑复代数簇族的覆盖与实数 o-极小扩张中局部可定义的结构类 $\mathcal{U}$ 相关联，证明了其子模型上的 $\aleph_0$-齐次性、稳定性及 $\aleph_1$-范畴性，并在一维情形下修改得到所有不可数基数下范畴的抽象初等类。

Characterizing categoricity in several classes of modules

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. $Theorem.$ Assume $R$ is an associative ring with unity. 1. The class of locally pure-injective $R$-modules is $λ$-categorical in $all$ $λ> |R|+\aleph_0$ if and only if $R \cong M_n(D)$ for $D$ a division ring and $n \geq 1$. 2. The class of flat $R$-modules is $λ$-categorical in $all$ $λ> |R| + \aleph_0$ if and only if $R \cong M_n(k)$ for $k$ a local ring such that its maximal ideal is left $T$-nilpotent and $n \geq 1$. 3. Assume $R$ is a commutative ring.…

Existentially closed De Morgan algebras

We show that the theory of De Morgan algebras has a model completion and axiomatise it. Then we prove that it is $\aleph_0$-categorical and describe definable and algebraic closures in that theory. We also obtain similar results for Boole-De Morgan algebras.

$2^{\aleph_0}$ pairwise non-isomorphic maximal-closed subgroups of Sym$(\mathbb{N})$ via the classification of the reducts of the Henson digraphs

Given two structures $\mathcal{M}$ and $\mathcal{N}$ on the same domain, we say that $\mathcal{N}$ is a reduct of $\mathcal{M}$ if all $\emptyset$-definable relations of $\mathcal{N}$ are $\emptyset$-definable in $\mathcal{M}$. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are $\aleph_0$-categorical, determining their reducts is equivalent to determining all closed supergroups $G<$ Sym$(\mathbb{N})$ of their automorphism groups. A consequence of the classific…

The Joint Embedding Property and Maximal Models

We introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(λ_i : i \le α<\aleph_1)$ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$(<λ_0)$, there is an $L_{ω_1,ω}$ -sentence $ψ$ whose models form a pure AEC and (1) The models of $ψ$ satisfy JEP$(<λ_0)$, while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist $2^{λ_i^+}$ non-isomorphic maximal mod…

Some stable non-elementary classes of modules

Fisher [Fis75] and Baur [Bau75] showed independently in the seventies that if $T$ is a complete first-order theory extending the theory of modules, then the class of models of $T$ with pure embeddings is stable. In [Maz4, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq_p)$ such that $K$ is a class of modules and $\leq_p$ is the pure submodule relation. In this paper we give some instances where this is true: $\textbf{Theorem.}$ Assume $R$ is an associative ring with unity. Let $(K, \leq_p)$ be an AEC such that $K \subseteq R\text{-Mod}$ and $K$ is closed unde…

Classification of $ω$-categorical monadically stable structures

A first-order structure $\mathfrak{A}$ is called monadically stable iff every expansion of $\mathfrak{A}$ by unary predicates is stable. In this article we give a classification of the class $\mathcal{M}$ of $ω$-categorical monadically stable structures in terms of their automorphism groups. We prove in turn that $\mathcal{M}$ is smallest class of structures which contains the one-element pure set, closed under isomorphisms, and closed under taking finitely disjoint unions, infinite copies, and finite index first-order reducts. Using our classification we show that every structure in $\mathcal…

Ax-Schanuel type theorems and geometry of strongly minimal sets in differentially closed fields

Let $(K;+,\cdot, ', 0, 1)$ be a differentially closed field. In this paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the geometry of the set $U:=\{ y:E(t,y) \wedge y' \neq 0 \}$ where $t$ is an element with $t'=1$. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of $U$. Moreover, the induced structure on Cartesian powers of $U$ is given by special subvarieties. If $E$ has some special form then all fibres $U_s:=\{ y:E(s,y) \wedge y' \neq 0 \}$ (with $s…