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Published: 10 December 2022
Last updated: 28 January 2023

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1

Vasey, Sebastien. "Quasiminimal abstract elementary classes." Archive for Mathematical Logic 57, no. 3-4 (June 28, 2017): 299–315. http://dx.doi.org/10.1007/s00153-017-0570-7.

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2

Hyttinen, Tapani. "Types in Abstract Elementary Classes." Notre Dame Journal of Formal Logic 45, no. 2 (April 2004): 99–108. http://dx.doi.org/10.1305/ndjfl/1095386646.

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3

Hirvonen, Åsa, and Tapani Hyttinen. "Metric abstract elementary classes with perturbations." Fundamenta Mathematicae 217, no. 2 (2012): 123–70. http://dx.doi.org/10.4064/fm217-2-2.

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4

Campion, Tim, and Jinhe Ye. "Homotopy types of abstract elementary classes." Journal of Pure and Applied Algebra 225, no. 5 (May 2021): 106461. http://dx.doi.org/10.1016/j.jpaa.2020.106461.

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5

Hyttinen, T., and M. Kesälä. "Independence in finitary abstract elementary classes." Annals of Pure and Applied Logic 143, no. 1-3 (November 2006): 103–38. http://dx.doi.org/10.1016/j.apal.2006.01.009.

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6

Kueker, David W. "Abstract elementary classes and infinitary logics." Annals of Pure and Applied Logic 156, no. 2-3 (December 2008): 274–86. http://dx.doi.org/10.1016/j.apal.2008.07.002.

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7

Beke, T., and J. Rosický. "Abstract elementary classes and accessible categories." Annals of Pure and Applied Logic 163, no. 12 (December 2012): 2008–17. http://dx.doi.org/10.1016/j.apal.2012.06.003.

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8

Shelah, Saharon, and Sebastien Vasey. "Abstract elementary classes stable in ℵ0." Annals of Pure and Applied Logic 169, no. 7 (July 2018): 565–87. http://dx.doi.org/10.1016/j.apal.2018.02.004.

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9

Lieberman, Michael, Jiří Rosický, and Sebastien Vasey. "Internal sizes in μ-abstract elementary classes." Journal of Pure and Applied Algebra 223, no. 10 (October 2019): 4560–82. http://dx.doi.org/10.1016/j.jpaa.2019.02.004.

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10

LIEBERMAN, M., and J. ROSICKÝ. "METRIC ABSTRACT ELEMENTARY CLASSES AS ACCESSIBLE CATEGORIES." Journal of Symbolic Logic 82, no. 3 (May 8, 2017): 1022–40. http://dx.doi.org/10.1017/jsl.2016.39.

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AbstractWe show that metric abstract elementary classes (mAECs) are, in the sense of [15], coherent accessible categories with directed colimits, with concrete ℵ1-directed colimits and concrete monomorphisms. More broadly, we define a notion of κ-concrete AEC—an AEC-like category in which only the κ-directed colimits need be concrete—and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah’s Presentation Theorem and a proof of the existence of an Ehrenfeucht–Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [15] yield a proof that any categorical mAEC is μ-d-stable in many cardinals below the categoricity cardinal.
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11

VanDieren, Monica M., and Sebastien Vasey. "Symmetry in abstract elementary classes with amalgamation." Archive for Mathematical Logic 56, no. 3-4 (March 28, 2017): 423–52. http://dx.doi.org/10.1007/s00153-017-0533-z.

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12

Boney, Will, Rami Grossberg, Michael Lieberman, Jiří Rosický, and Sebastien Vasey. "μ-Abstract elementary classes and other generalizations." Journal of Pure and Applied Algebra 220, no. 9 (September 2016): 3048–66. http://dx.doi.org/10.1016/j.jpaa.2016.02.002.

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13

GROSSBERG, RAMI, and MONICA VANDIEREN. "GALOIS-STABILITY FOR TAME ABSTRACT ELEMENTARY CLASSES." Journal of Mathematical Logic 06, no. 01 (June 2006): 25–48. http://dx.doi.org/10.1142/s0219061306000487.

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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 new context. We assume that [Formula: see text] is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include:. Theorem 0.1. Suppose that [Formula: see text] is not only tame, but [Formula: see text]-tame. If [Formula: see text] and [Formula: see text] is Galois stable in μ, then [Formula: see text], where [Formula: see text] is a relative of κ(T) from first order logic. [Formula: see text] is the Hanf number of the class [Formula: see text]. It is known that [Formula: see text]. The theorem generalizes a result from [17]. It is used to prove both the existence of Morley sequences for non-splitting (improving [22, Claim 4.15] and a result from [7]) and the following initial step towards a stability spectrum theorem for tame classes:. Theorem 0.2. If [Formula: see text] is Galois-stable in some [Formula: see text], then [Formula: see text] is stable in every κ with κμ=κ. For example, under GCH we have that [Formula: see text] Galois-stable in μ implies that [Formula: see text] is Galois-stable in μ+n for all n < ω.
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14

Lieberman, Michael J. "Category-theoretic aspects of abstract elementary classes." Annals of Pure and Applied Logic 162, no. 11 (November 2011): 903–15. http://dx.doi.org/10.1016/j.apal.2011.05.002.

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15

Jarden, Adi, and Saharon Shelah. "Non-forking frames in abstract elementary classes." Annals of Pure and Applied Logic 164, no. 3 (March 2013): 135–91. http://dx.doi.org/10.1016/j.apal.2012.09.007.

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16

Vasey, Sebastien. "Building independence relations in abstract elementary classes." Annals of Pure and Applied Logic 167, no. 11 (November 2016): 1029–92. http://dx.doi.org/10.1016/j.apal.2016.04.005.

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17

Boney, Will, Rami Grossberg, Monica M. VanDieren, and Sebastien Vasey. "Superstability from categoricity in abstract elementary classes." Annals of Pure and Applied Logic 168, no. 7 (July 2017): 1383–95. http://dx.doi.org/10.1016/j.apal.2017.01.005.

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18

Grossberg, Rami, and Marcos Mazari-Armida. "Simple-like independence relations in abstract elementary classes." Annals of Pure and Applied Logic 172, no. 7 (July 2021): 102971. http://dx.doi.org/10.1016/j.apal.2021.102971.

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19

Hyttinen, Tapani, and Meeri Kesälä. "Categoricity transfer in simple finitary abstract elementary classes." Journal of Symbolic Logic 76, no. 3 (September 2011): 759–806. http://dx.doi.org/10.2178/jsl/1309952520.

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AbstractWe continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak κ-categoricity and f-primary models to the framework of ℵ0-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let () be a simple finitary AEC, weakly categorical in some uncountable κ. Then () is weakly categorical in each λ ≥ min. If the class () is also -tame, weak κ-categoricity is equivalent with κ-categoricity in the usual sense.We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.
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20

Boney, Will, and Sebastien Vasey. "Structural logic and abstract elementary classes with intersections." Bulletin of the Polish Academy of Sciences Mathematics 67, no. 1 (2019): 1–17. http://dx.doi.org/10.4064/ba8178-12-2018.

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21

Friedman, Sy-David, and Martin Koerwien. "On Absoluteness of Categoricity in Abstract Elementary Classes." Notre Dame Journal of Formal Logic 52, no. 4 (July 2011): 395–402. http://dx.doi.org/10.1215/00294527-1499354.

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22

Boney, Will. "Advances in Classification Theory for Abstract Elementary Classes." Bulletin of Symbolic Logic 24, no. 4 (December 2018): 454–55. http://dx.doi.org/10.1017/bsl.2018.26.

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23

Baldwin, John, David Kueker, and Monica VanDieren. "Upward Stability Transfer for Tame Abstract Elementary Classes." Notre Dame Journal of Formal Logic 47, no. 2 (April 2006): 291–98. http://dx.doi.org/10.1305/ndjfl/1153858652.

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24

Lieberman, Michael, Jiří Rosický, and Sebastien Vasey. "Universal abstract elementary classes and locally multipresentable categories." Proceedings of the American Mathematical Society 147, no. 3 (December 6, 2018): 1283–98. http://dx.doi.org/10.1090/proc/14326.

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25

Boney, Will, and Rami Grossberg. "Forking in short and tame abstract elementary classes." Annals of Pure and Applied Logic 168, no. 8 (August 2017): 1517–51. http://dx.doi.org/10.1016/j.apal.2017.02.002.

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26

Grossberg, Rami, and Monica Vandieren. "Shelah's categoricity conjecture from a successor for tame abstract elementary classes." Journal of Symbolic Logic 71, no. 2 (June 2006): 553–68. http://dx.doi.org/10.2178/jsl/1146620158.

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AbstractWe prove a categoricity transfer theorem for tame abstract elementary classes.Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K+}. If K is categorical in λ and λ+, then K is categorical in λ++.Combining this theorem with some results from [37]. we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes:Suppose K is χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ0 ≔ Hanf(K). Ifand K is categorical in somethen K is categorical in μ for all μ .
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27

Hyttinen, Tapani, and Meeri Kesälä. "Lascar Types and Lascar Automorphisms in Abstract Elementary Classes." Notre Dame Journal of Formal Logic 52, no. 1 (January 2011): 39–54. http://dx.doi.org/10.1215/00294527-2010-035.

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28

GROSSBERG, RAMI, and SEBASTIEN VASEY. "EQUIVALENT DEFINITIONS OF SUPERSTABILITY IN TAME ABSTRACT ELEMENTARY CLASSES." Journal of Symbolic Logic 82, no. 4 (December 2017): 1387–408. http://dx.doi.org/10.1017/jsl.2017.21.

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AbstractIn the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:(1)For all high-enough λ,Khas no long splitting chains.(2)For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.(3)For all high-enough λ,Khas a unique limit model of cardinality λ.(4)For all high-enough λ,Khas a superlimit model of cardinality λ.(5)For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.(6)There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.
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29

HIRVONEN, ÅSA, and TAPANI HYTTINEN. "MEASURING DEPENDENCE IN METRIC ABSTRACT ELEMENTARY CLASSES WITH PERTURBATIONS." Journal of Symbolic Logic 82, no. 4 (December 2017): 1199–228. http://dx.doi.org/10.1017/jsl.2017.37.

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AbstractWe define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is dp-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we give a new example of such a class based on B. Zilber’s approximations of Weyl algebras. We introduce a way to measure the dependence of a tuple a from a set B over another set A. We prove basic properties of the notion, e.g., that a is independent of B over A in the usual sense of homogeneous model theory if and only if the measure of dependence is < ε for all ε > 0. In well behaved situations, the measure corresponds to the distance to a free extension. As an example of our measure of dependence we show a connection between the measure and entropy in models from quantum mechanics in which the spectrum of the observable is discrete. As an application, we show that weak simplicity implies a very strong form of simplicity and study the question of when the dependence inside a set of all realisations of some type can be seen to arise from a pregeometry in cases when the type is not regular. In the end of the paper, we demonstrate our notions and results in one more example: a class built from the p-adic integers.
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30

Vasey, Sebastien. "Saturation and solvability in abstract elementary classes with amalgamation." Archive for Mathematical Logic 56, no. 5-6 (May 27, 2017): 671–90. http://dx.doi.org/10.1007/s00153-017-0561-8.

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31

Vasey, Sebastien. "Toward a stability theory of tame abstract elementary classes." Journal of Mathematical Logic 18, no. 02 (November 20, 2018): 1850009. http://dx.doi.org/10.1142/s0219061318500095.

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We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior of saturated models.We deduce (in ZFC) that if a class is stable on a tail of cardinals, then it has no long splitting chains (the converse is known). This indicates that there is a clear notion of superstability in this framework.We also present an application to homogeneous model theory: for [Formula: see text] a homogeneous diagram in a first-order theory [Formula: see text], if [Formula: see text] is both stable in [Formula: see text] and categorical in [Formula: see text] then [Formula: see text] is stable in all [Formula: see text].
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32

Vasey, Sebastien. "Building prime models in fully good abstract elementary classes." Mathematical Logic Quarterly 63, no. 3-4 (October 11, 2017): 193–201. http://dx.doi.org/10.1002/malq.201600025.

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33

VanDieren, Monica. "Categoricity in abstract elementary classes with no maximal models." Annals of Pure and Applied Logic 141, no. 1-2 (August 2006): 108–47. http://dx.doi.org/10.1016/j.apal.2005.10.006.

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34

Baldwin, John T., and Olivier Lessmann. "Uncountable categoricity of local abstract elementary classes with amalgamation." Annals of Pure and Applied Logic 143, no. 1-3 (November 2006): 29–42. http://dx.doi.org/10.1016/j.apal.2006.01.007.

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35

Lieberman, Michael. "A topology for galois types in abstract elementary classes." Mathematical Logic Quarterly 57, no. 2 (March 1, 2011): 204–16. http://dx.doi.org/10.1002/malq.200910132.

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36

Johnson, Gregory M. "Abstract Elementary Classes with Löwenheim-Skolem Number Cofinal with ω." Notre Dame Journal of Formal Logic 51, no. 3 (July 2010): 361–71. http://dx.doi.org/10.1215/00294527-2010-022.

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37

Hyttinen, Tapani, and Gianluca Paolini. "Coxeter Groups and Abstract Elementary Classes: The Right-Angled Case." Notre Dame Journal of Formal Logic 60, no. 4 (November 2019): 707–31. http://dx.doi.org/10.1215/00294527-2019-0027.

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38

Hyttinen, Tapani. "Uncountably categorical local tame abstract elementary classes with disjoint amalgamation." Archive for Mathematical Logic 45, no. 1 (July 6, 2005): 63–73. http://dx.doi.org/10.1007/s00153-005-0305-z.

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39

GROSSBERG, RAMI, and MONICA VANDIEREN. "CATEGORICITY FROM ONE SUCCESSOR CARDINAL IN TAME ABSTRACT ELEMENTARY CLASSES." Journal of Mathematical Logic 06, no. 02 (December 2006): 181–201. http://dx.doi.org/10.1142/s0219061306000554.

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We prove that from categoricity in λ+ we can get categoricity in all cardinals ≥ λ+ in a χ-tame abstract elementary classe [Formula: see text] which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided [Formula: see text] and λ ≥ χ. For the missing case when [Formula: see text], we prove that [Formula: see text] is totally categorical provided that [Formula: see text] is categorical in [Formula: see text] and [Formula: see text].
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40

Villaveces, Andrés, and Pedro Zambrano. "Limit models in metric abstract elementary classes: the categorical case." Mathematical Logic Quarterly 62, no. 4-5 (August 2016): 319–34. http://dx.doi.org/10.1002/malq.201300060.

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41

Vasey, Sebastien. "On the uniqueness property of forking in abstract elementary classes." Mathematical Logic Quarterly 63, no. 6 (December 2017): 598–604. http://dx.doi.org/10.1002/malq.201700020.

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42

Kamsma, Mark. "Independence Relations in Abstract Elementary Categories." Bulletin of Symbolic Logic 28, no. 4 (December 2022): 531. http://dx.doi.org/10.1017/bsl.2022.27.

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AbstractIn model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example, linear independence in vector spaces yields such a relation.Some important classes in the stability hierarchy are stable, simple, and NSOP $_1$ , each being contained in the next. For each of these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example, simplicity is equivalent to admitting a certain independence relation, which must then be unique.All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic, and even a very general category-theoretic framework. In this thesis we continue this work.We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple, or NSOP $_1$ -like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing, and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.Switching frameworks, we generalise Kim-dividing in NSOP $_1$ theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP $_1$ if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.Abstract prepared by Mark Kamsma.E-mail:mark@markkamsma.nl.URL:https://markkamsma.nl/phd-thesis.
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43

Lieberman, Michael J. "Rank Functions and Partial Stability Spectra for Tame Abstract Elementary Classes." Notre Dame Journal of Formal Logic 54, no. 2 (2013): 153–66. http://dx.doi.org/10.1215/00294527-1960452.

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44

Zambrano, Pedro. "A stability transfer theorem in d -tame metric abstract elementary classes." Mathematical Logic Quarterly 58, no. 4-5 (July 20, 2012): 333–41. http://dx.doi.org/10.1002/malq.201110066.

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45

Vasey, Sebastien. "Shelah's eventual categoricity conjecture in tame abstract elementary classes with primes." Mathematical Logic Quarterly 64, no. 1-2 (April 2018): 25–36. http://dx.doi.org/10.1002/malq.201500068.

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46

Hyttinen, Tapani, and Gianluca Paolini. "Beyond abstract elementary classes: On the model theory of geometric lattices." Annals of Pure and Applied Logic 169, no. 2 (February 2018): 117–45. http://dx.doi.org/10.1016/j.apal.2017.10.003.

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47

Trlifaj, Jan. "Abstract elementary classes induced by tilting and cotilting modules have finite character." Proceedings of the American Mathematical Society 137, no. 03 (October 1, 2008): 1127–33. http://dx.doi.org/10.1090/s0002-9939-08-09618-4.

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48

VanDieren, M. M. "Symmetry and the union of saturated models in superstable abstract elementary classes." Annals of Pure and Applied Logic 167, no. 4 (April 2016): 395–407. http://dx.doi.org/10.1016/j.apal.2015.12.007.

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49

Lessmann, Olivier. "Upward categoricity from a successor cardinal for tame abstract classes with amalgamation." Journal of Symbolic Logic 70, no. 2 (June 2005): 639–60. http://dx.doi.org/10.2178/jsl/1120224733.

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AbstractThis paper is devoted to the proof of the following upward categoricity theorem: Let be a tame abstract elementary class with amalgamation, arbitrarily large models, and countable Löwenheim-Skolem number. If is categorical in ℵ then is categorical in every uncountable cardinal. More generally, wc prove that if is categorical in a successor cardinal λ+ then is categorical everywhere above λ+.
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50

BALDWIN, JOHN T., PAUL B. LARSON, and SAHARON SHELAH. "ALMOST GALOIS ω-STABLE CLASSES." Journal of Symbolic Logic 80, no. 3 (July 22, 2015): 763–84. http://dx.doi.org/10.1017/jsl.2015.19.

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AbstractTheorem. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presentable abstract elementary class with Löwenheim–Skolem number ℵ0, satisfying the joint embedding and amalgamation properties in ℵ0. If K has only countably many models in ℵ1, then all are small. If, in addition, k is almost Galois ω-stable then k is Galois ω-stable. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presented almost Galois ω-stable AEC satisfying amalgamation for countable models, and having a model of cardinality ℵ1. The assertion that K is ℵ1-categorical is then absolute.
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