arXiv subjects

Jeremy Beard

Publications and source records attributed to Jeremy Beard.

4 recordsLinked to original sources

arXiv2026

Disjoint non-forking amalgamation in stable AECs

The disjoint amalgamation property (DAP), which asserts that all spans of a class of models can be amalgamated with minimal intersection, is an important property in the context of abstract elementary classes, with connections to both Grossberg's question and Shelah's categoricity conjecture. We prove that, in a nice AEC $\mathbf{K}$ stable in $\lambda \geq \operatorname{LS}(\mathbf{K})$ with a strong enough independence relation, all high cofinality $\lambda$-limit models are disjoint (non-forking) amalgamation bases. $\textbf{Theorem.}$ Let $\mathbf{K}$ be an AEC stable in $\lambda$, where $\mathbf{K}_\lambda$ has AP, JEP, and NMM, and let $\mathbf{K}'$ be some AC where $\mathbf{K}_{(\lambda,\geq\kappa)} \subseteq \mathbf{K}' \subseteq \mathbf{K}_\lambda$. Suppose there is an independence relation on $\mathbf{K}'$ satisfying uniqueness, existence, non-forking amalgamation, $\mathbf{K}_{(\lambda,\geq\kappa)}$-universal continuity* in $\mathbf{K}_\lambda$, and $(\geq \kappa)$-local character. Assume $M_0, M_1, M_2 \in \mathbf{K}_{(\lambda,\geq\kappa)}$, and that $M_0 \leq_{\mathbf{K}} M_l$ and $a_l \in M_l$ for $l = 1, 2$. Then there exist $N \in \mathbf{K}_{(\lambda,\geq\kappa)}$ and $f_l : M_l \rightarrow N$ fixing $M_0$ for $l = 1, 2$ such that $\operatorname{gtp}(f_l(a_l)/f_{3-l}[M_{3-l}], N)$ does not fork over $M_0$ and $f_1[M_1] \cap f_2[M_2] = M_0$. That is, our independence relation has disjoint non-forking amalgamation in $\mathbf{K}_{(\lambda,\geq\kappa)}$. In particular, every $M_0 \in \mathbf{K}_{(\lambda,\geq\kappa)}$ is a disjoint amalgamation base in $\mathbf{K}_\lambda$. The hypotheses on the independence relation can be weakened (closer to $\lambda$-non-splitting in $\lambda$-stable AECs) if we are willing to give up the `non-forking' conditions of the amalgamation.

math.LO↗

arXiv2025

Long limit models are isomorphic assuming a splitting-like relation

We prove the uniqueness of high cofinality limit models in stable abstract elementary classes (AECs) with amalgamation, assuming the existence of a rather weak independence relation. $\textbf{Theorem.}$ Suppose $\mathbf{K}$ is a $\lambda$-stable AEC, where $\operatorname{LS}(\mathbf{K}) \leq \lambda$, $\kappa < \lambda^+$ is regular, and $\mathbf{K}_\lambda$ satisfies the amalgamation property. Let $\mathbf{K}'$ is the class of all $(\lambda, \delta)$-limit models where $\operatorname{cf}(\delta) \geq \kappa$ (or any AC where $\mathbf{K}' \subseteq \mathbf{K}_\lambda$ contains all such $(\lambda, \delta)$-limit models when $\operatorname{cf}(\delta) \geq \kappa$). Suppose also that there is an independence relation on $\mathbf{K}'$ satisfying weak uniqueness, weak existence, universal continuity* in $\mathbf{K}_\lambda$, $(\geq \kappa)$-local character, and $(\lambda, \theta)$-weak non-forking amalgamation in some regular $\theta \in [\kappa, \lambda^+)$. Let $\delta_1, \delta_2 < \lambda^+$ be limit with $\operatorname{cf}(\delta_l) \geq \kappa$ for $l = 1, 2$. Then for all $M, N_1, N_2 \in \mathbf{K}_\lambda$, if $N_l$ is $(\lambda, \delta_l)$-limit over $M$ for $l = 1, 2$, then $N_1 \underset{M}{\cong} N_2$. Moreover, if $K_\lambda$ also satisfies the joint embedding property, then for all $N_1, N_2 \in \mathbf{K}_\lambda$, if $N_l$ is $(\lambda, \delta_l)$-limit for $l = 1, 2$, then $N_1 {\cong} N_2$. This generalises both Theorem 3.1 of arXiv:2503.11605 and Theorem 1.2 of arXiv:1508.04717 - the former to apply to independence relations that satisfy much weaker forms of uniqueness, extension, and non-forking amalagamation, and the latter to independence relations other than $\lambda$-non-splitting. As such, this generalises all other positive isomorphism results of limit models known to the author.

math.LO↗

arXiv2025

The spectrum of limit models in a first order setting

Originally introduced by Kolmann and Shelah as a surrogate for saturated models, limit models have been established as natural and useful objects when studying abstract elementary classes. Shelah began the study of when (multiple notions of) limit models exist for first order theories. In this paper we look at their structure. In superstable theories it is known that all limit models are isomorphic, but in the strictly stable case the number of non-isomorphic limit models was not well understood. Here we characterise the full spectrum of limit models in the first order stable setting, by a short and simple argument using only the familiar machinery of stable first order theories: $\textbf{Theorem.}$ Let $T$ be a complete $\lambda$-stable theory where $\lambda \geq |\operatorname{L}(T)| + \aleph_0$. Let $\delta_1, \delta_2 < \lambda^+$ be limit ordinals where $\operatorname{cf}(\delta_1)< \operatorname{cf}(\delta_2)$. Let $N_l$ be a $(\lambda, \delta_l)$-limit model for $l = 1, 2$. Then $N_1$ and $N_2$ are isomorphic if and only if $\operatorname{cf}(\delta_1) \geq \kappa_r(T)$. Moreover, if $\kappa_r(T) = \aleph_\alpha$, there are exactly $|\alpha| + 1$ limit models up to isomorphism. In the context of first order stable theories, this reduces the proof of the main result of arXiv:2503.11605 from 19 pages to 2. We hope this will make limit models a more comprehensible and accessible tool in first order model theory.

math.LO↗

arXiv2025

On the spectrum of limit models

We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all `long' limit models are isomorphic, and all `short' limit models are non-isomorphic. $\textbf{Theorem.}$ Let $\mathbf{K}$ be a $\aleph_0$-tame abstract elementary class stable in $\lambda \geq \operatorname{LS}(\mathbf{K})$ with amalgamation, joint embedding and no maximal models. Suppose there is an independence relation on the models of size $\lambda$ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and $(\geq \kappa)$-local character in a minimal regular $\kappa < \lambda^+$. Suppose $\delta_1, \delta_2 < \lambda^+$ with $\operatorname{cf}(\delta_1) < \operatorname{cf}(\delta_2)$. Then for any $N_1, N_2, M \in \mathbf{K}_\lambda$ where $N_l$ is a $(\lambda, \delta_l)$-limit model over $M$ for $l = 1, 2$, \[N_1 \text{ is isomorphic to } N_2 \text{ over } M \iff \operatorname{cf}(\delta_1) \geq \kappa\] Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the $\aleph_0$-tameness assumption and assuming the independence relation is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation. We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes.

math.LO↗