No abstract

Abstract. We generalize Hrushovski's Group Configuration Theorem to quasiminimal classes. As an application, we present Zariski-like structures, a generalization of Zariski geometries, and show that a group can be found there if the pregeometry obtained from the bounded closure operator is non-trivial.Mathematics Subject Classification: 03C45, 03C48, 03C50, 03C98

Abstract.We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.

In this paper we study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in [Hy]. We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure, see [Hy]. Basic definitions and spectrum of stabilityThe purpose of this paper is to develop theory of independence for elementary submodels of a homogeneous structure. We get a model class of this kind if in addition to it's first-order theory we require that the models omit some (reasonable) set of types, see [Sh1]. If the set is empty, then we are in the 'classical situation' from [Sh2]. In other words, we study stability theory without the compactness theorem. So e.g. the theory of ∆ -ranks is lost and so we do not get an independence notion from ranks. Our independence notion is based on strong splitting. It satisfies the basic properties of forking in a rather weak form. The main problem is finding free extensions. So the arguments are often based on the definition of the independence notion instead of the 'independence-calculus'. * Partially supported by the Academi of Finland.† Research supported by the United States-Israel Binational Science Foundation. Publ. 629.Proof. (i), (ii) and (v) as in [Hy]. (iii) follows immediately from the homogeneity of M . (vi) is trivial.We prove (iv): Assume not. Let I be a counter example. Clearly we may assume that |I| = λ(M) . Then By Lemma 1.1, for every J ⊆ I , the typeProof. Follows immediately from Lemma 1.2 (v). We will use Lascar strong types instead of strong types:1.4 Definition. Let SE n (A) be the set of all equivalence relation E in M n , such that the number of equivalence classes is < |M| and for all f ∈ AutNotice that E ∈ SE(A) need not be definable but an indiscernible set over A is also an indiscernible set for all E ∈ SE(A) .Usually we either do not mention the arities of the equivalence relations we work with, or we mention that the arity is f.ex. m, but we do not specify what m is. This is harmless since usually there is no danger of confusion. 1.5 Lemma. If I is an infinite indiscernible set over A , then for all E ∈ SE(A) and a, b ∈ I , a E b .Proof. Assume not. Let E ∈ SE(A) be a counter example. Then for all a, b ∈ I , a = b , ¬(a E b) . Then Lemma 1.2 (iii) implies a contradiction with the number of equivalence classes of E . 1.6. Lemma. If E ∈ SE(A) , |A| ≤ ξ and M is ξ -stable, then the number of equivalence classes of E is ≤ ξ .Proof. Assume not. Then by Lemma 1.2 (ii), we can find I such that it is infinite indiscernible over A and for all a, b ∈ I , if a = b then ¬(a E b) . This contradicts Lemma 1.5.

We show that the excellence axiom in the definition of Zilber's quasiminimal excellent classes is redundant, in that it follows from the other axioms. This substantially simplifies a number of categoricity proofs.