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.

Upward categoricity from a successor cardinal for tame abstract classes with amalgamation

2005 ◽  
Vol 70 (2) ◽  
pp. 639-660 ◽  
Author(s):  
Olivier Lessmann
Keyword(s):  
Elementary Class ◽  
Uncountable Cardinal ◽  
Abstract Elementary Class

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+.

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.

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

A Purity Criterion for Pairs of Linear Transformations

1974 ◽  
Vol 26 (3) ◽  
pp. 734-745 ◽  
Author(s):  
Uri Fixman ◽  
Frank A. Zorzitto
Keyword(s):  
Perturbation Methods ◽  
Eigenvalue Problems ◽  
Vector Spaces ◽  
Infinite Dimension ◽  
Linear Transformations ◽  
Complex Vector ◽  
Algebraic Investigation ◽  
Image Position

In connection with the study of perturbation methods for differential eigenvalue problems, Aronszajn put forth a theory of systems (X, Y; A, B) consisting of a pair of linear transformations A, B:X → Y (see [1]; cf. also [2]). Here X and Y are complex vector spaces, possibly of infinite dimension. The algebraic aspects of this theory, where no restrictions of topological nature are imposed, where developed in [3] and [5]. We hasten to point out that the category of C2-systems (definition in § 1) in which this algebraic investigation takes place is equivalent to the category of all right modules over the ring of matrices of the form

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

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.

A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals

2006 ◽  
Vol 136 (4) ◽  
pp. 701-708 ◽  
Author(s):  
Jonathan Bevan ◽  
Pablo Pedregal
Keyword(s):  
Lower Semicontinuous ◽  
Short Note ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
Lower Semicontinuous Envelope ◽  
Variational Integrals ◽  
Non Local ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Bounded Below

In this short note we prove that the functional I : W1,p(J;R) → R defined by is sequentially weakly lower semicontinuous in W1,p(J,R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull Wsc.

scholarly journals Saturated Ideals in Boolean Extensions

1972 ◽  
Vol 48 ◽  
pp. 159-168 ◽  
Author(s):  
Yuzuru Kakuda
Keyword(s):  
Regular Cardinal ◽  
Uncountable Cardinal

0. Introduction. Let k be an uncountable cardinal, and let λ be a regular cardinal less than k. Let I be a λ-saturated non-trivial ideal on k. Prikry, in his thesis, showed that, in certain Boolean extensions, k has a λ-saturated non-trivial ideal on k.

scholarly journals CONSTRUCTING TYPES IN DIFFERENTIALLY CLOSED FIELDS THAT ARE ANALYSABLE IN THE CONSTANTS

2018 ◽  
Vol 83 (04) ◽  
pp. 1413-1433
Author(s):  
RUIZHANG JIN
Keyword(s):  
Minimal Length ◽  
Canonical Analysis ◽  
Base Property ◽  
Canonical Base ◽  
Closed Fields ◽  
Image Position ◽  
Positive Integers

AbstractAnalysability of finiteU-rank types are explored both in general and in the theory${\rm{DC}}{{\rm{F}}_0}$. The well-known fact that the equation$\delta \left( {{\rm{log}}\,\delta x} \right) = 0$is analysable in but not almost internal to the constants is generalized to show that$\underbrace {{\rm{log}}\,\delta \cdots {\rm{log}}\,\delta }_nx = 0$is not analysable in the constants in$\left( {n - 1} \right)$-steps. The notion of acanonical analysisis introduced–-namely an analysis that is of minimal length and interalgebraic with every other analysis of that length. Not every analysable type admits a canonical analysis. Using properties of reductions and coreductions in theories with the canonical base property, it is constructed, for any sequence of positive integers$\left( {{n_1}, \ldots ,{n_\ell }} \right)$, a type in${\rm{DC}}{{\rm{F}}_0}$that admits a canonical analysis with the property that theith step hasU-rank${n_i}$.

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