Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let m be a cardinal above the the Löwenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit models of cardinality m. If K is m-Galois-stable, has no m-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, for the precise description of long splitting chains and locality}, then any two (m,sigma_i)-limits over M for (i in {1,2}) are isomorphic over M.
This theorem extends results of Shelah, Kolman and Shelah, and Shelah and Villaveces. A preliminary version of our uniqueness theorem was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes.
We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in previous work of Baldwin, Kolesnikov, and Shelah. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of Baldwin, Kolesnikov, and Shelah by showing, in ZFC, that the (disjoint) amalgamation property for classes $K_α$ studied in that paper must hold up to $\beth_α$ (only a…
We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum-sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.
We prove: Main Theorem: Let $\mathcal{K}$ be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality $μ$. Let $μ$ be a cardinal above the the Löwenheim-Skolem number of the class. If $\mathcal{K}$ is $μ$-Galois-stable, has no $μ$-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two $(μ,σ_\ell)$-limits over $M$, for $\ell\in\{1,2\}$, are isomorphic over $M$. This theorem extends results of Shelah from \cite{Sh394}, \cite{Sh576}, \cite{Sh600}, Kolman and Shelah in \cite{Ko…
Let K be an abstract elementary class with amalgamation, and Lowenheim Skolem number LS(K). We prove that for a suitable Hanf number chi_0 if chi_0 < lambda_0 <= lambda_1, and K is categorical in lambda^+_1 then it is categorical in lambda_0 .
We combine two approaches to the study of classification theory of AECs: 1. that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and 2. that of Grossberg and VanDieren: (studying non-splitting) assuming the amalgamation property and tameness. In [JrSh875], we derive a good non-forking $λ^+$-frame from a semi-good non-forking $λ$-frame. But the classes $K_{λ^+}$ and $\preceq \restriction K_{λ^+}$ are replaced: $K_{λ^+}$ is restricted to the saturated models and the partial order $\preceq \restriction K_{λ^+}$ is …
This paper continues the study of generalized amalgamation properties. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and link the binding group of the groupoids to a certain automorphism group of the monster model, showing that the group must be abelian as well. We also study connections between n-existence and n-uniqueness properties for various "dimensions" n in the wider context of simple theories. We introduce a family of weaker existence and uniqueness properties. Many of…
We provide a self-contained introduction to the classical theory of universal-homogeneous models (also known as generic structures, rich models, or Fraïssé limits). In the literature, most treatments restrict consideration to embeddings among finite structures. This is not suitable for some applications. We take the notion of morphisms as primitive and we allow structures to have arbitrary cardinality.
We develop some basic results about full amalgamation classes with intrinsic trascendentals. These classes have generics whose models may have finite subsets whose intrinsic closure is not contained in its algebraic closure. We will show that under fairly natural conditions the generic will have an essentially undecidable theory, but we will also exhibit strictly superstable and strictly simple examples. Separating types over a model into those that are intrinsic and those that are extrinsic, we will demonstrate that the complexity exceeding that of a simple theory in the classes with essentia…
$\mathbf{Theorem.}$ Let $K$ be an abstract elementary class (AEC) with amalgamation and no maximal models. Let $λ> \text{LS} (K)$. If $K$ is categorical in $λ$, then the model of cardinality $λ$ is Galois-saturated. This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: $K$ has a unique limit model in each cardinal below $λ$, (when $λ$ is big-enough) $K$ is weakly tame below $λ$, and the thresholds of several existing categoricity transfers can be improved. We also prove a downward transfer of solvability (a version of superstability introduced by She…
Let $K_d$ denote the class of all finite graphs and, for graphs $A\subseteq B$, say $A \leq_d B$ if distances in $A$ are preserved in $B$; i.e. for $a, a' \in A$ the length of the shortest path in $A$ from $a$ to $a'$ is the same as the length of the shortest path in $B$ from $a$ to $a'$. In this situation $(K_d, \leq_d)$ forms an amalgamation class and one can perform a Hrushovski construction to obtain a generic of the class. One particular feature of the class $(K_d, \leq_d)$ is that a closed superset of a finite set need not include all minimal pairs obtained iteratively over that set but …
We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure $M$, Polish group $G$ of permutations of $M$, and $n \geq 1$, $G$ has a comeager $n$-diagonal conjugacy class iff the family of all $n$-tuples of $G$-extendable bijections between finitely generated substructures of $M$, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé classes that are not homogenizable. Finally, we investigate $1$- and $2$-diagonal conjugacy cl…
This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-elementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolated by the first author) in AECs with amalgamation that satisfy a local definition of superstability. The key results are a downward transfer of symmetry and a deduction of symmetry from failure of the o…
Generalizing and simplifying recent work of Dobrinen, we show that if $\mathcal{L}$ is a finite binary relational language and $\mathcal{F}$ is a finite set of finite irreducible $\mathcal{L}$-structures, then the class $\mathcal{K} = \mathrm{Forb}(\mathcal{F})$ has finite big Ramsey degrees.
For $K$ an abstract elementary class with amalgamation and no maximal models, we show that categoricity in a high-enough cardinal implies structural properties such as the uniqueness of limit models and the existence of good frames. This improves several classical results of Shelah. $\mathbf{Theorem}$ Let $μ\ge \text{LS} (K)$. If $K$ is categorical in a $λ\ge \beth_{\left(2^μ\right)^+}$, then: 1) Whenever $M_0, M_1, M_2 \in K_μ$ are such that $M_1$ and $M_2$ are limit over $M_0$, we have $M_1 \cong_{M_0} M_2$. 2) If $μ> \text{LS} (K)$, the model of size $λ$ is $μ$-saturated. 3) If $μ\ge \beth_…
The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The long-term goal is to solve Shelah's Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by proving that in their context, the uniqueness of limit models follows from categoricity under the assumption that the subclass of amalgamation bases is closed under unions of bounded, increasing chains.
Motivated by the free products of groups, the direct sums of modules, and Shelah's $(λ,2)$-goodness, we study strong amalgamation properties in Abstract Elementary Classes. Such a notion of amalgamation consists of a selection of certain amalgams for every triple $M_0\leq M_1, M_2$, and we show that if $K$ designates a unique strong amalgam to every triple $M_0\leq M_1, M_2$, then $K$ satisfies categoricity transfer at cardinals $\geqθ(K)+2^{\text{LS}(K)}$, where $θ(K)$ is a cardinal associated with the notion of amalgamation. We also show that if such a unique choice does not exist, then ther…
Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding property, let μbe the Hanf number of K. Suppose K is tame. MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than \beth_{(2^μ)^+} then K is categorical in all cardinals greater than \beth_{(2^μ)^+}. This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a strongly compact cardinal for the same conclusion) and Shelah's downward categoricity theorem for AECs with amalgamation (from [Sh394]).
The class of linearly ordered sets with one order preserving unary operation has the Strong Amalgamation Property (SAP). The class of linearly ordered sets with one strict order preserving unary operation has AP but not SAP. The class of linearly ordered sets with two order preserving unary operations has not AP. For every set $F$, the class of linearly ordered sets with an $F$-indexed family of automorphisms has SAP. Corresponding results are proved in the case of order reversing operations. Various subclasses of the above classes are considered and some model-theoretical consequences are pre…
The assumption that an AEC is tame is a powerful assumption permitting development of stability theory for AECs with the amalgamation property. Lately several upward categoricity theorems were discovered where tameness replaces strong set-theoretic assumptions. We present in this article two sufficient conditions for tameness, both in form of strong amalgamation properties that occur in nature. One of them was used recently to prove that several Hrushovski classes are tame. This is done by introducing the property of weak $(μ,n)$-uniqueness which makes sense for all AECs (unlike Shelah's origi…
We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not assume amalgamation, and do not use large cardinals. Moreover we give an explicit bound on the "high-enough" threshold: $\mathbf{Theorem}$ Let $ψ$ be a universal $\mathbb{L}_{ω_1, ω}$ sentence. If $ψ$ is categorical in some $λ\ge \beth_{\beth_{ω_1}}$, then $ψ$ is categorical in all $λ' \ge \beth_{\beth_{ω_1}}$. As a byproduct of the proof, we show that a conj…
We study classes of graded structures satisfying the properties of amalgamation, joint embedding and hereditariness. Given appropriate conditions, we can build a graded analogue of the Fraisse limit. Some examples such as the class of all finite weighted graphs or the class of all finite fuzzy orders (evaluated on a particular countable algebra) will be examined.
We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements. $\mathbf{Theorem}$ (Superstability from categoricity) Let $K$ be a $(<κ)$-tame AEC with amalgamation. If $κ= \beth_κ> \text{LS} (K)$ and $K$ is categorical in a $λ> κ$, then: * $K$ is stable in all cardinals $\ge κ$. * $K$ is categorical in $κ$. * There is a type-full good $λ$-frame with underlying class $K…
We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on t…
In this paper we examine the task set forth by Shelah and Villaveces in \cite{ShVi} of proving the uniqueness of limit models of cardinality $μ$ in $λ$-categorical abstract elementary classes with no maximal models, where $λ$ is some cardinal larger than $μ$. In \cite{Va} and \cite{Va-errata} we identified several gaps in the approach outlined in \cite{ShVi}, and we added the assumption that the union of an increasing chain of limit models is a limit model. Here we replace this assumption with the seemingly weaker statement that the union of an increasing and continuous chain of limit models i…
We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which…
Let K be an abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties. Theorem 1. Suppose K is χ-tame. If K is categorical in some λ^+ >LS(K) then it is categorical in all μ\geq (λ+χ)^+. Theorem 2. If K is LS(K)-tame and is categorical both in LS(K) and in LS(K)^+ then K is categorical in all μ\geq LS(K).
Our main result (Theorem 1) suggests a possible dividing line ($μ$-superstable $+$ $μ$-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of such a dividing line. Theoerem 1: Let $\mathcal{K}$ be an abstract elementary class with no maximal models of cardinality $μ^+$ which satisfies the joint embedding and amalgamation properties. Suppose $μ\geq LS(\mathcal{K})$. If $\mathcal{K}$ is $μ$- and $μ^+$-superstable and satisfies $μ^+$-symmetry, then for any increasing sequence $\langle M_i\in\mathcal{K}_…
We provide here the first steps toward Classification Theory of Abstract Elementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some lambda greater than its Lowenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non mu-splitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the Los Conjecture for these classes.
This paper is concerned with a class K of models and an abstract notion of submodel <=. Experience in first order model theory has shown the desirability of finding a `monster model' to serve as a universal domain for K. In the original constructions of Jonsson and Fraisse, K was a universal class and ordinary substructure played the role of <=. Working with a cardinal lambda satisfying lambda^{< lambda}= lambda guarantees appropriate downward Lowenheim-Skolem theorems; the existence and uniqueness of a homogeneous-universal model appears to depend centrally on the amalgamation property. We ma…
In this paper we prove: Theorem 1. Let $\mathcal{K}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $λ>μ\geq LS(\mathcal{K})$ and $θ$ is a limit ordinal $<λ^+$. If $\mathcal{K}$ is $μ$ superstable and $μ^+$-superstable and satisfies $μ^+$-symmetry, then for any increasing sequence $\langle M_i\mid i<θ\rangle$ of $μ^+$-saturated models of cardinality $λ$, the model $\bigcup_{i<θ}M_i$ is $μ^+$-saturated.
We study abstract elementary classes (AECs) that, in $\aleph_0$, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at $\aleph_0$. More precisely, there is a superlimit model of cardinality $\aleph_0$ and the class generated by this superlimit has a type-full good $\aleph_0$-frame (a local notion of nonforking independence) and a superlimit model of cardinality $\aleph_1$. We also give a supersimplicity condition under which the local…
We study notions of independence appropriate for a stability theory of metric abstract elementary classes (for short, MAECs). We build on previous notions used in the discrete case, and adapt definitions to the metric case. In particular, we study notions that behave well under superstability-like assumptions. Also, under uniqueness of limit models, we study domination, orthogonality and parallelism of Galois types in MAECs.
We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this context. We assume that $\K$ is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: (1) Galois-stability above the Hanf number implies that κ(K) is less than the Hanf number. Where κ(K) is the parallel of \kapppa(T) for f.o. T. (2) We use (1) to construct Morley sequences (for non-splitting) improving previo…
We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Nešetřil-Rödl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results s…
In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for $μ$-forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking-like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent.
In an effort to find more examples of amalgamation classes whose almost sure theories are the same as their generic theories as well as amalgamation classes whose almost sure theories are different from their generic theories, we address our attention to two new cases: the bipartite diameter 3 metrically homogeneous graphs of generic type. These graphs were identified by Cherlin, and are determined by certain forbidden configurations. In this paper, we explicitly identify and enumerate their ages, for which we then establish both unlabeled and labeled 0-1 laws. Finally, we show that for one of…
We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle.
Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called nonsplitting. We generalize their result as follows: given an abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the Shelah-Villaveces result (the independence notion there was splitting) as…
The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum hypothesis, GCH), we give a complete list of the possible categoricity spectrums of an abstract elementary class with amalgamation and arbitrarily large models. Specifically, the categoricity spectrum is either empty, an end segment starting below the Hanf number, or a closed interval consisting of finite successors of the Löwenheim-Skolem-Tarski number (there are e…
We study some amalgamation classes introduced by Cherlin and prove the simplicity of the automorphism groups of the Fra{\"ı}ss{é} limits of these classes. We employ the machinery of stationary independence relations used by Tent and Ziegler.
In this paper we address a problem posed by Shelah in 1999 to find a suitable notion for superstability for abstract elementary classes in which limit models of cardinality $μ$ are saturated. Theorem 1. Suppose that $\mathcal{K}$ is a $χ$-tame abstract elementary class with no maximal models satisfying the joint embedding property and the amalgamation property. Suppose $μ$ is a cardinal with $μ\geq\beth_{(2^{LS(\mathcal{K})+χ})^+}$. Let $M$ be a model of cardinality $μ$. If $\mathcal{K}$ is both $χ$-stable and $μ$-stable and satisfies the $μ$-superstability assumptions, then any two $μ$-limit …
This paper continues the study of superstability in abstract elementary classes (AECs) satisfying the amalgamation property. In particular, we consider the definition of $μ$-superstability which is based on the local character characterization of superstability from first order logic. Not only is $μ$-superstability a potential dividing line in the classification theory for AECs, but it is also a tool in proving instances of Shelah's Categoricity Conjecture. In this paper, we introduce a formulation, involving towers, of symmetry over limit models for $μ$-superstable abstract elementary classes…
