DISJOINT AMALGAMATION IN LOCALLY FINITE AEC

2017 ◽  
Vol 82 (1) ◽  
pp. 98-119 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
MARTIN KOERWIEN ◽  
MICHAEL C. LASKOWSKI
Keyword(s):  
Locally Finite ◽  
Elementary Class ◽  
Image Position ◽  
Abstract Elementary Class

AbstractWe 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 complete sentence in ${L_{{\omega _1},\omega }}$ where the spectrum of cardinals in which amalgamation holds is other that none or all.

AN ABSTRACT ELEMENTARY CLASS NONAXIOMATIZABLE IN

2019 ◽  
Vol 84 (3) ◽  
pp. 1240-1251
Author(s):  
SIMON HENRY
Keyword(s):  
Regular Cardinal ◽  
Vector Spaces ◽  
Left Adjoint ◽  
Scott Topology ◽  
Elementary Class ◽  
Closed Fields ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Bounded Below ◽  
Abstract Elementary Class

AbstractWe show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a ${L_{\infty ,\omega }}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of κ-points of a κ-topos, in particular, nor the category of models of a ${L_{\infty ,\kappa }}$-theory.The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree.

scholarly journals ALMOST GALOIS ω-STABLE CLASSES

2015 ◽  
Vol 80 (3) ◽  
pp. 763-784 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
PAUL B. LARSON ◽  
SAHARON SHELAH
Keyword(s):  
Elementary Class ◽  
Joint Embedding ◽  
Image Position ◽  
Abstract Elementary Class ◽  
Countable Models

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.

Existentially closed central extensions of locally finite p-groups

1986 ◽  
Vol 100 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Felix Leinen ◽  
Richard E. Phillips
Keyword(s):  
Central Extensions ◽  
Locally Finite ◽  
Existentially Closed ◽  
Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

scholarly journals N⊥ as an abstract elementary class

2007 ◽  
Vol 149 (1-3) ◽  
pp. 25-39 ◽  
Author(s):  
John T. Baldwin ◽  
Paul C. Eklof ◽  
Jan Trlifaj
Keyword(s):  
Elementary Class ◽  
Abstract Elementary Class

Independence, dimension and continuity in non-forking frames

2013 ◽  
Vol 78 (2) ◽  
pp. 602-632 ◽  
Author(s):  
Adi Jarden ◽  
Alon Sitton
Keyword(s):  
Continuity Property ◽  
Natural Conditions ◽  
Elementary Class ◽  
Independence Dimension ◽  
The Stability ◽  
Abstract Elementary Class

AbstractThe notion J is independent in (M, M0, N) was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal λ and has a non-forking relation, satisfying the good λ-frame axioms and some additional hypotheses. Shelah uses independence to define dimension.Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved.As a corollary, we weaken the stability hypothesis and two additional hypotheses, that appear in Shelah's theorem.

STONE SPACE OF CYLINDRIC ALGEBRAS AND TOPOLOGICAL MODEL SPACES

2016 ◽  
Vol 81 (3) ◽  
pp. 1069-1086
Author(s):  
CHARLES C. PINTER
Keyword(s):  
Boolean Algebra ◽  
Model Theory ◽  
Representation Theorem ◽  
Stone Space ◽  
Equivalence Relations ◽  
Cylindric Algebra ◽  
Cylindric Algebras ◽  
Locally Finite ◽  
Abstract Model Theory ◽  
Image Position

AbstractThe Stone representation theorem was a milestone for the understanding of Boolean algebras. From Stone’s theorem, every Boolean algebra is representable as a field of sets with a topological structure. By means of this, the structural elements of any Boolean algebra, as well as the relations between them, are represented geometrically and can be clearly visualized. It is no different for cylindric algebras: Suppose that ${\frak A}$ is a cylindric algebra and ${\cal S}$ is the Stone space of its Boolean part. (Among the elements of the Boolean part are the diagonal elements.) It is known that with nothing more than a family of equivalence relations on ${\cal S}$ to represent quantifiers, ${\cal S}$ represents the full cylindric structure just as the Stone space alone represents the Boolean structure. ${\cal S}$ with this structure is called a cylindric space.Many assertions about cylindric algebras can be stated in terms of elementary topological properties of ${\cal S}$. Moreover, points of ${\cal S}$ may be construed as models, and on that construal ${\cal S}$ is called a model space. Certain relations between points on this space turn out to be morphisms between models, and the space of models with these relations hints at the possibility of an “abstract” model theory. With these ideas, a point-set version of model theory is proposed, in the spirit of pointless topology or category theory, in which the central insight is to treat the semantic objects (models) homologously with the corresponding syntactic objects so they reside together in the same space.It is shown that there is a new, purely algebraic way of introducing constants in cylindric algebras, leading to a simplified proof of the representation theorem for locally finite cylindric algebras. Simple rich algebras emerge as homomorphic images of cylindric algebras. The topological version of this theorem is especially interesting: The Stone space of every locally finite cylindric algebra ${\frak A}$ can be partitioned into subspaces which are the Stone spaces of all the simple rich homomorphic images of ${\frak A}$. Each of these images completely determines a model of ${\frak A}$, and all denumerable models of ${\frak A}$ appear in this representation.The Stone space ${\cal S}$ of every cylindric algebra can likewise be partitioned into closed sets which are duals of all the types in ${\frak A}$. This fact yields new insights into miscellaneous results in the model theory of saturated models.

scholarly journals Shelah's categoricity conjecture from a successor for tame abstract elementary classes

2006 ◽  
Vol 71 (2) ◽  
pp. 553-568 ◽  
Author(s):  
Rami Grossberg ◽  
Monica Vandieren
Keyword(s):  
Abstract Elementary Classes ◽  
Transfer Theorem ◽  
Elementary Class ◽  
Joint Embedding ◽  
Elementary Classes ◽  
Abstract Elementary Class

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 .

scholarly journals FORKING AND SUPERSTABILITY IN TAME AECS

2016 ◽  
Vol 81 (1) ◽  
pp. 357-383 ◽  
Author(s):  
SEBASTIEN VASEY
Keyword(s):  
General Construction ◽  
Transfer Theorem ◽  
Elementary Class ◽  
Abstract Elementary Class ◽  
Limit Models

AbstractWe prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models.

HANDMADE DENSITY SETS

2017 ◽  
Vol 82 (1) ◽  
pp. 208-223 ◽  
Author(s):  
GEMMA CAROTENUTO
Keyword(s):  
Set Theory ◽  
Real Line ◽  
Borel Measure ◽  
Point Of View ◽  
Topological Complexity ◽  
Descriptive Set Theory ◽  
Locally Finite ◽  
Finite Borel Measure ◽  
Image Position ◽  
Descriptive Set

AbstractGiven a metric space (X , d), equipped with a locally finite Borel measure, a measurable set $A \subseteq X$ is a density set if the points where A has density 1 are exactly the points of A. We study the topological complexity of the density sets of the real line with Lebesgue measure, with the tools—and from the point of view—of descriptive set theory. In this context a density set is always in $\Pi _3^0$. We single out a family of true $\Pi _3^0$ density sets, an example of true $\Sigma _2^0$ density set and finally one of true $\Pi _2^0$ density set.

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