Limit models in metric abstract elementary classes: the categorical case

Villaveces, Andrés; Zambrano, Pedro

doi:10.1002/malq.201300060

Cited by 9 publications

(18 citation statements)

References 16 publications

(54 reference statements)

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“…The question of uniqueness is intriguing when the cofinalities of the lengths of the chains are different. This question has been studied in many papers, among them [ShVi99], [Van06], [GVV16], [Bon14a], [Van16], [BoVan], [ViZa16] and [Vas].…”

Section: Preliminariesmentioning

confidence: 99%

“…The key question has been the uniqueness of limit models of the same cardinality but with chains of different lengths. This has been studied thoroughly [ShVi99], [Van06], [GVV16], [Bon14a], [Van16], [BoVan], [ViZa16] and [Vas]. In this same line, [GrVas17] and [Vas16] showed that if a class has a monster model and is tame then uniqueness of limit models on a tail of cardinals is equivalent to being Galois-superstable 3 .…”

Section: Introductionmentioning

confidence: 99%

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Algebraic description of limit models in classes of abelian groups

Mazari-Armida

2020

Annals of Pure and Applied Logic

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We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation. We show:Theorem 0.1.(1) If G is a limit model of cardinality λ in the class of abelian groups with the subgroup relation, then G ∼ = (⊕ λ Q) ⊕ ⊕ p prime (⊕ λ Z(p ∞ )).(2) If G is a limit model of cardinality λ in the class of torsion-free abelian groups with the pure subgroup relation, then:• If the length of the chain has uncountable cofinality, then• If the length of the chain has countable cofinality, then G is not algebraically compact.We also study the class of finitely Butler groups with the pure subgroup relation, we show that it is an AEC, Galois-stable and (< ℵ 0 )-tame and short.2 For a more detailed introduction to the theory of AECs we suggest the reader to look at [Gro02], [Bal09] or [BoVas17] (this only covers tame AECs, but the AECs that we will study in this paper are all tame). 3 We say that K is Galois-superstable if there is µ < (2 LS(K) ) + such that K is λ-Galois-stable for every λ ≥ µ.Under the assumption of joint embedding, amalgamation, no maximal models and LS(K)-tameness (which hold for all the classes studied in this paper, except perhaps the one introduced in the last section) by [GrVas17] and [Vas18] the definition of the previous line is equivalent to any other definition of Galois-superstability given in the context of AECs. 4Recall that H is a pure subgroup of G if for every n ∈ N it holds that nG ∩ H = nH.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Algebraic description of limit models in classes of abelian groups

Mazari-Armida

2020

Annals of Pure and Applied Logic

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“…As mentioned above, we work in the context of metric AECs (mAECs), as considered in [10], [21], and [22]: classes in which the structures have complete metric spaces rather than sets as their sorts, and where the interpretations of the function and relation symbols are required to behave well with respect to the appropriate metrics. To be precise, let L be a language with sorts S ∪ {R}, function symbols F ∪ {d σ } σ∈S , relation symbols R, and constant symbols C.…”

Section: Metric Aecsmentioning

confidence: 99%

Metric Abstract Elementary Classes as Accessible Categories

Lieberman¹,

Rosický²

2017

J. symb. log.

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We show that metric abstract elementary classes (mAECs) are, in the sense of [16], coherent accessible categories with directed colimits, with concrete ℵ 1directed 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 categorytheoretic 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 [16] yield a proof that any categorical mAEC is µ-d-stable in many cardinals below the categoricity cardinal. 2 M. LIEBERMAN AND J. ROSICKÝ(4) (K, U) is a coherent accessible category with concrete directed colimits and concrete monomorphisms, where "coherence" is a property of U corresponding to the coherence axiom for AECs. (5) (K, U) is a coherent accessible category with concrete directed colimits and concrete monomorphisms, and satisfies the iso-fullness condition described in Remark 3.5 in [16]-such a category is equivalent to an AEC.Certain essential results from the theory of AECs are shown to hold at greater levels of generality: categories of the form (2) satisfy a presentation theorem generalizing that of Shelah and, if large, admit a robust EM-functor. Categories of the form (3) allow the development of Galois types and satisfy a generalization of Boney's theorem on tameness under the assumption of a proper class of strongly compact cardinals (see [7]). Categories of the form (4) satisfy the essential technical condition that Galois saturation corresponds to, in AEC terms, model-homogeneity, and support the development of a fragment of classification theory. We show in Section 3 that any mAEC K is an accessible category with directed colimits, and note that, if we take U : K → Set to be the usual underlying set functor, K is coherent with concrete monomorphisms. As is well known, though, directed colimits in K need not be concrete: when taking the colimit of a chain of structures in K, we must, in general, take the completion of the union of the underlying sets. This would seem to place us, at best, in type (2) above, which is already sufficient to give a presentation theorem and guarantee the existence of an EM-functor for a general mAEC-this is in itself a generalization of [10], the results of which hold only in the homogenous case. As we note in Remark 2.9, however, mAECs do have concrete ℵ 1 -directed colimits, which suggests that we may benefit from a generalization of the hierarchy of [16], considering categories with concrete κ-directed colimits for some κ. In case a category of this form has (not necessarily concrete) directed colimits, is coherent, has concrete monomorphisms, and is suitably replete and iso-full-the conditions of (4) above-we call it a κ-concrete AEC, or κ-CAEC for short. Incidentally, there is an alternative option already being pursued in, e.g., [8], namely to consider classes of structu...

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“…Then there is a unique map π from the bounded Borel functions on R into B(H ) such that Theorem 2. 26 Let Q be a closed unbouned self-adjoint operator. Then, for every λ ∈ R, the following conditions are equivalent:…”

Section: Definition 217mentioning

confidence: 99%

“…In order to deal with the case of analytic structures, Hyttinen and Hirvonen defined metric abstract elementary classes in as a generalization of Shelah's AECs to classes of metric structures (MAECs). After this, Villaveces and Zambrano studied notions of independence and superstability for metric abstract elementary clases (MAECs) .…”

Section: Introductionmentioning

confidence: 99%

Model theory of a Hilbert space expanded with an unbounded closed selfadjoint operator

Pulido

2014

Math. Log. Quart.

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We study a closed unbounded self‐adjoint operator Q acting on a Hilbert space H in the framework of Metric Abstract Elementary Classes (MAECs). We build a suitable MAEC for such a structure, prove it is ℵ0‐categorical and ℵ0‐stable up to a system of perturbations. We give an explicit continuous Lω1,ω axiomatization for the class. We also characterize non‐splitting and show it has the same properties as non‐forking in superstable first order theories. Finally, we characterize equality, orthogonality and domination of (Galois) types in that MAEC.

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Limit models in metric abstract elementary classes: the categorical case

Abstract: We study versions of limit models adapted to the context of metric abstract elementary classes. Under categoricity and superstability‐like assumptions, we generalize some theorems from . We prove criteria for existence and uniqueness of limit models in the metric context.

Cited by 9 publications

References 16 publications

Algebraic description of limit models in classes of abelian groups

Algebraic description of limit models in classes of abelian groups

Metric Abstract Elementary Classes as Accessible Categories

Model theory of a Hilbert space expanded with an unbounded closed selfadjoint operator

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