We describe the major result on categoricity in Lω1,ω , placing it in the context of more general work in abstract elementary classes. In particular, we illustrate the role of higher dimensional amalgamations and sketch the role of a weak extension of ZFC in the proof. We expound the translation of the problem to studying atomic models of first order theories. We provide a simple example of the...

We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals. Theorem A. For every natural number k, there is a class Kk defined by a sentence in Lω1,ω that has no models of cardinality greater than ik+1, but Kk has the disjoint amalgamation property on models of cardinality less than or equal to אk−3 and has models of cardinality אk−1. More strong...

Theorem. For each 2 ≤ k < ω there is an Lω1,ω-sentence φk such that: (1) φk is categorical in μ if μ ≤ אk−2; (2) φk is not אk−2-Galois stable; (3) φk is not categorical in any μ with μ > אk−2; (4) φk has the disjoint amalgamation property; (5) For k > 2, (a) φk is (א0,אk−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most אk−3; (b) φk is אm-Gal...

We give an explicit description of the homology group Hn(p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups Hi(q) are trivial for 2 ≤ i < n. The group Hn(p) turns out to be isomorphic to the automorphism group of a certain piece of the algebraic closure of n independent realizations of p; it was shown earlier by the authors tha...

We present a property of neat reducts commuting with forming subalgebras as a definability condition. The purpose of this paper is to relate results on neat reducts, a notion particular to cylindric algebras to results on the “more universal” strong amalgamation property in a very general setting. It is known [7], [8] and [6] that amalgamation properties in a class of algebras correspond to int...

We prove that for each ,nite core graph G, the class of all graphs admitting a homomorphism into G is a pseudo-amalgamation class, in the sense of Fra01ss2 e. This fact gives rise to a countably in,nite universal pseudo-homogeneous graph which shares some of the properties of the in,nite random graph. Our methods apply simultaneously to G-colourings in several classes of relational structures, ...

Associated to each elliptic root system, we introduce a Lie algebra as an amalgamation of an a ne Kac-Moody algebra and a generalization of a Heisenberg algebra. The derived algebra of the amalgamation algebra is shown to be the universal central extension of a certain twisted loop algebra of two variables.

Let $f: Arightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we investigate the transfer of the property of coherence to the amalgamation $Abowtie^{f}J$. We provide necessary and sufficient conditions for $Abowtie^{f}J$ to be a coherent ring.

Deductive object-oriented databases advocate the advantage of combining object-oriented and deductive paradigms into a single data model. Certainly, the query language in such a data model has to reeect the amalgamation because it works as the interface to the user and/or application program. This paper proposes a language to formulate queries as classes related to the schema classes and constr...