scholarly journals THE HANF NUMBER FOR AMALGAMATION OF COLORING CLASSES

2016 ◽  
Vol 81 (2) ◽  
pp. 570-583 ◽  
Author(s):  
ALEXEI KOLESNIKOV ◽  
CHRIS LAMBIE-HANSON
Keyword(s):  
Amalgamation Property ◽  
Abstract Elementary Classes ◽  
The Family ◽  
Hanf Number ◽  
Elementary Classes

AbstractWe study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in [3]. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of [3] by showing, in ZFC, that the (disjoint) amalgamation property for classes Kα studied in that paper must hold up to ℶα (only a consistency result was previously known).

scholarly journals GALOIS-STABILITY FOR TAME ABSTRACT ELEMENTARY CLASSES

2006 ◽  
Vol 06 (01) ◽  
pp. 25-48 ◽  
Author(s):  
RAMI GROSSBERG ◽  
MONICA VANDIEREN
Keyword(s):  
Initial Step ◽  
Amalgamation Property ◽  
Order Logic ◽  
Abstract Elementary Classes ◽  
First Order ◽  
Elementary Class ◽  
Hanf Number ◽  
Elementary Classes ◽  
Abstract Elementary Class ◽  
Stability Results

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.

Examples of non-locality

2008 ◽  
Vol 73 (3) ◽  
pp. 765-782 ◽  
Author(s):  
John T. Baldwin ◽  
Saharon Shelah
Keyword(s):  
Abelian Groups ◽  
Amalgamation Property ◽  
Abstract Elementary Classes ◽  
Elementary Classes ◽  
Positive Effect ◽  
Non Locality

AbstractWe use κ-free but not Whitehead Abelian groups to construct Abstract Elementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galois-types. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC's which preserves non-locality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is an AEC with amalgamation which is not (ℵ0, ℵ1)-tame but is (, ∞)-tame; Theorem 3.3. It is consistent with ZFC that there is an AEC with amalgamation which is not (≤ ℵ2, ≤ ℵ2)-compact.

scholarly journals Independence in finitary abstract elementary classes

2006 ◽  
Vol 143 (1-3) ◽  
pp. 103-138 ◽  
Author(s):  
T. Hyttinen ◽  
M. Kesälä
Keyword(s):  
Abstract Elementary Classes ◽  
Elementary Classes
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