On universal modules with pure embeddings

We show that certain classes of modules have universal models with respect to pure embeddings. $Theorem.$ Let $R$ be a ring, $T$ a first-order theory with an infinite model extending the theory of $R$-modules and $K^T=(Mod(T), \leq_{pp})$ (where $\leq_{pp}$ stands for pure submodule). Assume $K^T$ has joint embedding and amalgamation. If $\lambda^{|T|}=\lambda$ or $\forall \mu < \lambda( \mu^{|T|} < \lambda)$, then $K^T$ has a universal model of cardinality $\lambda$. As a special case we get a recent result of Shelah [Sh17, 1.2] concerning the existence of universal reduced torsion-free abelian groups with respect to pure embeddings. We begin the study of limit models for classes of $R$-modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure-injective and we characterize limit models with chains of countable cofinality. This can be used to answer Question 4.25 of [Maz]. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.Comment: 17 page

Hanf Numbers and Presentation Theorems in AECs

We prove that a strongly compact cardinal is an upper bound for a Hanf number for amalgamation, etc. in AECs using both semantic and syntactic methods. To syntactically prove non-disjoint amalgamation, a different presentation theorem than Shelah's is needed. This relational presentation theorem has the added advantage of being {\it functorial}, which allows the transfer of amalgamation

Forking and superstability in tame AECs

We prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models.Comment: 33 page

The categoricity spectrum of large abstract elementary classes

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 L\"owenheim-Skolem-Tarski number (there are examples of each type). We also prove (assuming a strengthening of the GCH) that the categoricity spectrum of an abstract elementary class with no maximal models is either bounded or contains an end segment. This answers several longstanding questions around Shelah's categoricity conjecture.Comment: 50 page

A model theoretic solution to a problem of L\'{a}szl\'{o} Fuchs

Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $\lambda$ such that there is a universal abelian $p$-group for purity of cardinality $\lambda$, i.e., an abelian $p$-group $U_\lambda$ of cardinality $\lambda$ such that every abelian $p$-group of cardinality $\leq \lambda$ purely embeds in $U_\lambda$. In this paper we use ideas from the theory of abstract elementary classes to show: $\textbf{Theorem.}$ Let $p$ be a prime number. If $\lambda^{\aleph_0}=\lambda$ or $\forall \mu < \lambda( \mu^{\aleph_0} < \lambda)$, then there is a universal abelian $p$-group for purity of cardinality $\lambda$. Moreover for $n\geq 2$, there is a universal abelian $p$-group for purity of cardinality $\aleph_n$ if and only if $2^{\aleph_0} \leq \aleph_n$. As the theory of abstract elementary classes has barely been used to tackle algebraic questions, an effort was made to introduce this theory from an algebraic perspective.Comment: 10 page

Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes

Our main result (Theorem 1) suggests a possible dividing line ($\mu$-superstable $+$ $\mu$-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 $\mu^+$ which satisfies the joint embedding and amalgamation properties. Suppose $\mu\geq LS(\mathcal{K})$. If $\mathcal{K}$ is $\mu$- and $\mu^+$-superstable and satisfies $\mu^+$-symmetry, then for any increasing sequence $\langle M_i\in\mathcal{K}_{\geq\mu^{+}}\mid i<\theta<(\sup\|M_i\|)^+\rangle$ of $\mu^+$-saturated models, $\bigcup_{i<\theta}M_i$ is $\mu^+$-saturated. We also apply results of VanDieren's Superstability and Symmetry paper and use towers to transfer symmetry from $\mu^+$ down to $\mu$ in abstract elementary classes which are both $\mu$- and $\mu^+$-superstable: Theorem 2: Suppose $\mathcal{K}$ is an abstract elementary class satisfying the amalgamation and joint embedding properties and that $\mathcal{K}$ is both $\mu$- and $\mu^+$-superstable. If $\mathcal{K}$ has symmetry for non-$\mu^+$-splitting, then $\mathcal{K}$ has symmetry for non-$\mu$-splitting.Comment: This paper is a synthesis of arXiv:1507.01991 and arXiv:1507.0198

A survey on tame abstract elementary classes

Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification theory for them have been made. Abstract independence relations such as Shelah's good frames have been found to be key objects. Several new categoricity transfers have been obtained. We survey these developments using the following result (due to the second author) as our guiding thread: $\mathbf{Theorem}$ If a universal class is categorical in cardinals of arbitrarily high cofinality, then it is categorical on a tail of cardinals.Comment: 84 page

Joint diamonds and Laver diamonds

The concept of jointness for guessing principles, specifically $\diamondsuit_\kappa$ and various Laver diamonds, is introduced. A family of guessing sequences is joint if the elements of any given sequence of targets may be simultaneously guessed by the members of the family. While equivalent in the case of $\diamondsuit_\kappa$, joint Laver diamonds are nontrivial new objects. We give equiconsistency results for most of the large cardinals under consideration and prove sharp separations between joint Laver diamonds of different lengths in the case of $\theta$-supercompact cardinals.Comment: 34 pages; revised version with several improvements, including expanded Sections 3.3 and

Tameness, Uniqueness and amalgamation

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 $\lambda^+$-frame from a semi-good non-forking $\lambda$-frame. But the classes $K_{\lambda^+}$ and $\preceq \restriction K_{\lambda^+}$ are replaced: $K_{\lambda^+}$ is restricted to the saturated models and the partial order $\preceq \restriction K_{\lambda^+}$ is restricted to the partial order $\preceq^{NF}_{\lambda^+}$. Here, we avoid the restriction of the partial order $\preceq \restriction K_{\lambda^+}$, assuming that every saturated model (in $\lambda^+$ over $\lambda$) is an amalgamation base and $(\lambda,\lambda^+)$-tameness for non-forking types over saturated models, (in addition to the hypotheses of [JrSh875]): We prove that $M \preceq M^+$ if and only if $M \preceq^{NF}_{\lambda^+}M^+$, provided that $M$ and $M^+$ are saturated models. We present sufficient conditions for three good non-forking $\lambda^+$-frames: one relates to all the models of cardinality $\lambda^+$ and the two others relate to the saturated models only. By an `unproven claim' of Shelah, if we can repeat this procedure $\omega$ times, namely, `derive' good non-forking $\lambda^{+n}$ frame for each $n<\omega$ then the categoricity conjecture holds. Vasey applies one of our main theorems in a proof of the categoricity conjecture under the above `unproven claim' of Shelah and more assumptions. In [Jrprime], we apply the main theorem in a proof of the existence of primeness triples

Good Frames With A Weak Stability

Let K be an abstract elementary class of models. Assume that there are less than the maximal number of models in K_{\lambda^{+n}} (namely models in K of power \lambda^{+n}) for all n. We provide conditions on K_\lambda, that imply the existence of a model in K_{\lambda^{+n}} for all n. We do this by providing sufficiently strong conditions on K_\lambda, that they are inherited by a properly chosen subclass of K_{\lambda^+}