# DISJOINT AMALGAMATION IN LOCALLY FINITE AEC

@article{Baldwin2017DISJOINTAI,
title={DISJOINT AMALGAMATION IN LOCALLY FINITE AEC},
author={John T. Baldwin and Martin Koerwien and Michael C. Laskowski},
journal={The Journal of Symbolic Logic},
year={2017},
volume={82},
pages={98 - 119}
}
• Published 1 March 2017
• Computer Science, Mathematics
• The Journal of Symbolic Logic
Abstract We introduce the concept of a locally finite abstract elementary class and develop the theory of disjoint $\left( { \le \lambda ,k} \right)$ -amalgamation) for such classes. From this we find a family of complete ${L_{{\omega _1},\omega }}$ sentences ${\phi _r}$ that a) homogeneously characterizes ${\aleph _r}$ (improving results of Hjorth [11] and Laskowski–Shelah [13] and answering a question of [21]), while b) the ${\phi _r}$ provide the first examples of a class of models of a…
15 Citations

## Topics from this paper

Disjoint n-Amalgamation and Pseudofinite Countably Categorical Theories
• Alex Kruckman
• Mathematics, Computer Science
Notre Dame J. Formal Log.
• 2019
This paper shows that if a countably categorical theory T admits an expansion with disjoint $n-amalgamation for all$n$then T is pseudofinite, and two generic theories of equivalence relations are examined, and it is shown that both are pseud ofinite. Shelah’s eventual categoricity conjecture in universal classes: part II AbstractWe prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the Shelah's eventual categoricity conjecture in universal classes: Part I • S. Vasey • Mathematics, Computer Science Ann. Pure Appl. Log. • 2017 A Lower Bound for the Hanf Number for Joint Embedding. • Mathematics • 2018 In [13] the authors show that if$\mu$is a strongly compact cardinal,$K$is an Abstract Elementary Class (AEC) with$LS(K)<\mu$, and$K$satisfies joint embedding (amalgamation) cofinally below Higher-dimensional obstructions for star reductions • Mathematics Fundamenta Mathematicae • 2021 A$*$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a$*$-reduction between Amalgamating many overlapping Boolean algebras In general, two overlapping Boolean algebras always extend to a common Boolean algebra, but three may not. We prove a new sufficient condition for$n$overlapping Boolean algebras to have a common Non-absoluteness of Hjorth's Cardinal Characterization • Mathematics • 2021 In [5], Hjorth proved that for every countable ordinal α, there exists a complete Lω1,ω-sentence φα that has models of all cardinalities less than or equal to אα, but no models of cardinality אα+1. Characterizing the existence of a Borel complete expansion • Mathematics • 2021 We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence Φ as a class of structures in a related language. From this, we show that Φ has a Borel A survey on tame abstract elementary classes • Mathematics • 2015 Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification ## References SHOWING 1-10 OF 50 REFERENCES Characterizing the powerset by a complete (Scott) sentence This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing the work from this http URL A cardinal$\kappa$is characterized by a Scott sentence$\phi_M$, The joint embedding property and maximal models • Mathematics, Computer Science Arch. Math. Log. • 2016 It is proved the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence and is shown that although AP(\kappa )$$AP(κ) for each$$\kappa $$κ implies the full amalgamation property, JEP(kappa) forEach does not imply the full joint embedding property. Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence The set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products, and it is proved that if \aleph_\alpha is characterized byA Scott sentence, at least one of \ aleph_alpha and is homogeneously characterizable. Categoricity, amalgamation, and tameness • Mathematics • 2009 AbstractTheorem: For each 2 ≤ k < ω there is an$$ L_{\omega _1 ,\omega }$$-sentence ϕk such that(1) ϕk is categorical in μ if μ≤ℵk−2;(2) ϕk is not ℵk−2-Galois stable(3) ϕk is not categorical in Complete$\mathcal{L}_{\omega_1,\omega}\$-Sentences with Maximal Models in Multiple Cardinalities
• Mathematics
• 2015
In [BKS15] examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this
The amalgamation spectrum
• Computer Science, Mathematics
The Journal of Symbolic Logic
• 2009
It is shown that the ℵ∝ to ℶ∝ in the second theorem consistently with ZFC and while havingℵi ≪ ℷi for 0 < i < ∝ and similar results hold for arbitrary ordinals ∝.
Classification theory for non-elementary classes I: The number of uncountable models ofψ ∈Lω_1, ω. Part A
Assuming that 2Nn < 2Nn+1 forn < ω, we prove that everyψ ∈Lω_1, ω has many non-isomorphic models of powerNn for somen>0or has models in all cardinalities. We can conclude that every such Ψ has at
On the Existence of Atomic Models
• Mathematics, Computer Science
J. Symb. Log.
• 1993
Knight, Kueker and Shelah independently showed that the converse holds, provided that the cardinality of the underlying language has size at most @1.
Classification Theory for Abstract Elementary Classes
An abstract elementary class is a class of structures of the same vocabulary (like a class of rings, or a class of fields), with a partial order that generalizes the relation "A is a substructure (or
Three red herrings around Vaught’s conjecture
• Mathematics
• 2015
We give a model theoretic proof that if there is a counterexample to Vaught’s conjecture there is a counterexample such that every model of cardinality א1 is maximal (strengthening a result of