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Monica M. Vandieren - One of the best experts on this subject based on the ideXlab platform.
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Symmetry in abstract Elementary Classes with amalgamation
Archive for Mathematical Logic, 2017Co-Authors: Monica M. Vandieren, Sebastien VaseyAbstract: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 order property. These results are then used to prove several structural properties in categorical AECs, improving Classical results of Shelah who focused on the special case of categoricity in a successor cardinal. We also study the interaction of symmetry with tameness, a locality property for Galois (orbital) types. We show that superstability and tameness together imply symmetry. This sharpens previous work of Boney and the second author.
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Superstability from categoricity in abstract Elementary Classes
Annals of Pure and Applied Logic, 2017Co-Authors: Will Boney, Rami Grossberg, Monica M. Vandieren, Sebastien VaseyAbstract:Abstract 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 nonsplitting, a particular notion of independence. We generalize their result as follows: given any 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 well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah–Villaveces proof.
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Symmetry and the union of saturated models in superstable abstract Elementary Classes
Annals of Pure and Applied Logic, 2016Co-Authors: Monica M. VandierenAbstract:Abstract 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. Theorem 1 Let K be an abstract Elementary Class with no maximal models of cardinality μ + which satisfies the joint embedding and amalgamation properties. Suppose μ ≥ LS ( K ) . If K is μ- and μ + -superstable and satisfies μ + -symmetry, then for any increasing sequence 〈 M i ∈ K ≥ μ + | i θ ( sup ‖ M i ‖ ) + 〉 of μ + -saturated models, ⋃ i θ M i is μ + -saturated. We also apply results of [18] and use towers to transfer symmetry from μ + down to μ in abstract Elementary Classes which are both μ- and μ + -superstable: Theorem 2 Suppose K is an abstract Elementary Class satisfying the amalgamation and joint embedding properties and that K is both μ- and μ + -superstable. If K has symmetry for non- μ + -splitting, then K has symmetry for non-μ-splitting.
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On the structure of categorical abstract Elementary Classes with amalgamation
arXiv: Logic, 2015Co-Authors: Monica M. Vandieren, Sebastien VaseyAbstract: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 $\mu \ge \text{LS} (K)$. If $K$ is categorical in a $\lambda \ge \beth_{\left(2^{\mu}\right)^+}$, then: 1) Whenever $M_0, M_1, M_2 \in K_\mu$ are such that $M_1$ and $M_2$ are limit over $M_0$, we have $M_1 \cong_{M_0} M_2$. 2) If $\mu > \text{LS} (K)$, the model of size $\lambda$ is $\mu$-saturated. 3) If $\mu \ge \beth_{(2^{\text{LS} (K)})^+}$ and $\lambda \ge \beth_{\left(2^{\mu^+}\right)^+}$, then there exists a type-full good $\mu$-frame with underlying Class the saturated models in $K_\mu$. Our main tool is the symmetry property of splitting (previously isolated by the first author). The key lemma deduces symmetry from failure of the order property.
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Limit Models in Strictly Stable Abstract Elementary Classes
arXiv: Logic, 2015Co-Authors: Will Boney, Monica M. VandierenAbstract:In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract Elementary Classes. In particular we prove: Suppose that $K$ is an abstract Elementary Class satisfying 1. the joint embedding and amalgamation properties with no maximal model of cardinality $\mu$. 2. stabilty in $\mu$. 3. $\kappa_{\mu}(K)
Sebastien Vasey - One of the best experts on this subject based on the ideXlab platform.
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The categoricity spectrum of large abstract Elementary Classes
Selecta Mathematica, 2019Co-Authors: Sebastien VaseyAbstract: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 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.
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Quasiminimal abstract Elementary Classes
Archive for Mathematical Logic, 2017Co-Authors: Sebastien VaseyAbstract:We propose the notion of a quasiminimal abstract Elementary Class (AEC). This is an AEC satisfying four semantic conditions: countable L\"owenheim-Skolem-Tarski number, existence of a prime model, closure under intersections, and uniqueness of the generic orbital type over every countable model. We exhibit a correspondence between Zilber's quasiminimal pregeometry Classes and quasiminimal AECs: any quasiminimal pregeometry Class induces a quasiminimal AEC (this was known), and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry Class. We show in particular that the exchange axiom is redundant in Zilber's definition of a quasiminimal pregeometry Class.
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Saturation and solvability in abstract Elementary Classes with amalgamation
Archive for Mathematical Logic, 2017Co-Authors: Sebastien VaseyAbstract:Theorem 0.1 Let \(\mathbf {K}\) be an abstract Elementary Class (AEC) with amalgamation and no maximal models. Let \(\lambda > {LS}(\mathbf {K})\). If \(\mathbf {K}\) is categorical in \(\lambda \), then the model of cardinality \(\lambda \) is Galois-saturated.
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Symmetry in abstract Elementary Classes with amalgamation
Archive for Mathematical Logic, 2017Co-Authors: Monica M. Vandieren, Sebastien VaseyAbstract: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 order property. These results are then used to prove several structural properties in categorical AECs, improving Classical results of Shelah who focused on the special case of categoricity in a successor cardinal. We also study the interaction of symmetry with tameness, a locality property for Galois (orbital) types. We show that superstability and tameness together imply symmetry. This sharpens previous work of Boney and the second author.
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Superstability from categoricity in abstract Elementary Classes
Annals of Pure and Applied Logic, 2017Co-Authors: Will Boney, Rami Grossberg, Monica M. Vandieren, Sebastien VaseyAbstract:Abstract 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 nonsplitting, a particular notion of independence. We generalize their result as follows: given any 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 well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah–Villaveces proof.
Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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Infinitary Logics and Abstract Elementary Classes.
arXiv: Logic, 2021Co-Authors: Saharon Shelah, Andrés VillavecesAbstract:We prove that every abstract Elementary Class (a.e.c.) with LST number $\kappa$ and vocabulary $\tau$ of cardinality $\leq \kappa$ can be axiomatized in the logic ${\mathbb L}_{\beth_2(\kappa)^{+++},\kappa^+}(\tau)$. In this logic an a.e.c. is therefore an EC Class rather than merely a PC Class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the \emph{canonical tree} $\mathcal S={\mathcal S}_{\mathcal K}$ of an a.e.c. $\mathcal K$. This turns out to be an interesting combinatorial object of the Class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary logic $L^1_\lambda$.}
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Classification Theory for Abstract Elementary Classes
2009Co-Authors: Saharon ShelahAbstract: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 an Elementary substructure) of B". The requirements are that the Class is closed under isomorphism, and that isomorphic structures have isomorphic (generalized) substructures; we also require that our Classes share some of the most basic properties of Elementary Classes, like closure under unions of increasing chains of substructures. We would like to Classify this general family; in the sense of proving dichotomies: either we can understand the structure of all models in our Class or there are many to some extent. More specifically we would like to generalize the theory about categoricity and superstability to this context.
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Introduction to: Classification theory for abstract Elementary Class
arXiv: Logic, 2009Co-Authors: Saharon ShelahAbstract:Classification theory of Elementary Classes deals with first order (Elementary) Classes of structures (i.e. fixing a set T of first order sentences, we investigate the Class of models of T with the Elementary submodel notion). It tries to find dividing lines, prove their consequences, prove "structure theorems, positive theorems" on those in the "low side" (in particular stable and superstable theories), and prove "non-structure, complexity theorems" on the "high side". It has started with categoricity and number of non-isomorphic models. It is probably recognized as the central part of model theory, however it will be even better to have such (non-trivial) theory for non-Elementary Classes. Note also that many Classes of structures considered in algebra are not first order; some families of such Classes are close to first order (say have kind of compactness). But here we shall deal with a Classification theory for the more general case without assuming knowledge of the first order case (and in most parts not assuming knowledge of model theory at all). The present paper includes an introduction to the forthcoming book on Classification Theory for Abstract Elementary Classes
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Karp complexity and Classes with the independence property
arXiv: Logic, 2003Co-Authors: Michael C. Laskowski, Saharon ShelahAbstract:A Class K of structures is controlled if for all cardinals lambda, the relation of L_{infty,lambda}-equivalence partitions K into a set of equivalence Classes (as opposed to a proper Class). We prove that no pseudo-Elementary Class with the independence property is controlled. By contrast, there is a pseudo-Elementary Class with the strict order property that is controlled.
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Karp complexity and Classes with the independence property
Annals of Pure and Applied Logic, 2003Co-Authors: Michael C. Laskowski, Saharon ShelahAbstract:A Class K of structures is controlled if for all cardinals λ, the relation of L∞,λ-equivalence partitions K into a set of equivalence Classes (as opposed to a proper Class). We prove that no pseudo-Elementary Class with the independence property is controlled. By contrast, there is a pseudo-Elementary Class with the strict order property that is controlled (see Arch. Math. Logic 40 (2001) 69–88).
Rami Grossberg - One of the best experts on this subject based on the ideXlab platform.
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Superstability from categoricity in abstract Elementary Classes
Annals of Pure and Applied Logic, 2017Co-Authors: Will Boney, Rami Grossberg, Monica M. Vandieren, Sebastien VaseyAbstract:Abstract 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 nonsplitting, a particular notion of independence. We generalize their result as follows: given any 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 well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah–Villaveces proof.
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Superstability in abstract Elementary Classes
arXiv: Logic, 2015Co-Authors: Rami Grossberg, Sebastien VaseyAbstract:In the context of abstract Elementary Class (AEC) with amalgamation, joint embedding, and arbitrarily large models, an AEC is \emph{$\lambda$-superstable} if it is stable in $\lambda$ and has no long splitting chains in $\lambda$. Under the assumptions that the Class is tame and stable, we prove that several other definitions of superstability are equivalent in this context. This partially answers questions of Shelah. $\mathbf{Theorem}$ Let $K$ be a tame AEC with amalgamation, joint embedding, and arbitrarily large models. Assume that $K$ is stable in a proper Class of cardinals. Then the following are equivalent: 1) For all high-enough $\lambda$, $K$ is $\lambda$-superstable. 2) For all high-enough $\lambda$, there exists a good $\lambda$-frame on a skeleton of $K_\lambda$. 3) For all high-enough $\lambda$, $K$ has a unique limit model of cardinality $\lambda$. 4) For all high-enough $\lambda$, $K$ has a superlimit model of cardinality $\lambda$. 5) For all high-enough $\lambda$, the union of any increasing chain of $\lambda$-saturated models is $\lambda$-saturated. 6) There exists $\mu$ such that for all high-enough $\lambda$, $K$ is $(\lambda, \mu)$-solvable.
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Galois-stability for Tame abstract Elementary Classes
Journal of Mathematical Logic, 2006Co-Authors: Rami Grossberg, Monica M. VandierenAbstract: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 is a tame abstract Elementary Class satisfying the amalgamation property with no maximal model. The main results include:. Theorem 0.1. Suppose that is not only tame, but -tame. If and is Galois stable in μ, then , where is a relative of κ(T) from first order logic. is the Hanf number of the Class . It is known that . 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 is Galois-stable in some , then is stable in every κ with κμ=κ. For example, under GCH we have that Galois-stable in μ implies that is Galois-stable in μ+n for all n < ω.
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Shelah's Categoricity Conjecture from a Successor for Tame Abstract Elementary Classes
Journal of Symbolic Logic, 2006Co-Authors: Rami Grossberg, Monica M. VandierenAbstract:We 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 μ 0 ≔ Hanf( K ). If and K is categorical in some then K is categorical in μ for all μ .
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GALOIS-STABILITY FOR TAME ABSTRACT Elementary ClassES
Journal of Mathematical Logic, 2006Co-Authors: Rami Grossberg, Monica M. VandierenAbstract: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.
Ann V. Angell - One of the best experts on this subject based on the ideXlab platform.
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Practicing Democracy at School: A Qualitative Analysis of an Elementary Class Council
Theory & Research in Social Education, 1998Co-Authors: Ann V. AngellAbstract:Abstract Advocates of democratic education argue that regular Class meetings are essential to the school curriculum, offering students practice in democratic process as they deliberate issues that affect them. This article describes an experiment with regular Class meetings over three years in a mixed-age upper Elementary Class. Students readily adopted the rudiments of parliamentary order and also invented democratic procedures to achieve their goals. Analysis of the minutes of 216 meetings suggested students' implicit goals were self-definition and consensus-building; explicitly they defended respect, fairness, and the right to work undisturbed. Negotiating standards for conduct, sharing information, and planning events provided opportunities for students to improve deliberation skills, develop empathy, and build community. The mixed ages in the Class appeared to facilitate the development of moral reasoning. Students' inclination to imitate peers, however, suggests their need for help developing tolera...