We consider the question of when, given a subset A of M, the setwise stabilizer of the group of automorphisms induces a closed subgroup on Sym(A). We define s-homogeneity to be the analogue of homogeneity relative to strong embeddings and show that any subset of a countable, s-homogeneous, ω-stable structure induces a closed subgroup and contrast this with a number of negative results. We also show that for ω-stable structures s-homogeneity is preserved under naming countably many constants, but under slightly (...) weaker conditions it can be lost by naming a single point. (shrink)

The paper continues [1]. Let S be a complete theory of ultraflat (e.g. planar) graphs as introduced in [4]. We show a strong form of NOTOP for S: The union of two models M1 and M2, independent over a common elementary submodel M0, is the primary model over M1 ∪ M2 of S. Then by results of [1] Mekler's construction [6] gives for such a theory S of nice ultraflat graphs a superstable 2-step-nilpotent group of exponent $p (>2)$ with NDOP (...) and NOTOP. (shrink)

We study the expansion of stable structures by adding predicates for arbitrary subsets. Generalizing work of Poizat-Bouscaren on the one hand and Baldwin-Benedikt-Casanovas-Ziegler on the other we provide a sufficient condition for such an expansion to be stable. This generalization weakens the original definitions in two ways: dealing with arbitrary subsets rather than just submodels and removing the ‘small' or ‘belles paires' hypothesis. We use this generalization to characterize in terms of pairs, the ‘triviality' of the geometry on a strongly (...) minimal set. Call a set A benign if any type over A in the expanded language is determined by its restriction to the base language. We characterize the notion of benign as a kind of local homogenity. Answering a question of [8] we characterize the property that M has the finite cover property over A. (shrink)

The main results in the paper are the following. Theorem A. Suppose that T is superstable and M ⊂ N are distinct models of T eq . Then there is a c ϵ N⧹M such that t is regular. For M ⊂ N two models we say that M ⊂ na N if for all a ϵ M and θ such that θ ≠ θ , there is a b ∈ θ ⧹ acl . Theorem B Suppose that T is (...) superstable , M ⊂ na N are models of T eq , and p is a regular type non-orthogonal to t . Then there is a c ϵ N such that t is regular and non-orthogonal to p. Furthermore, there is a formula θ ∈ t such that a ∈ θ and t ⊥ ̷ p ⇒ t is regular. We used these results to obtain ‘good’ tree decompositions of models in superstable theories with NDOP. See Definition 5.1 for the undefined terms. Theorem C Suppose that T is superstable with NDOP and M ⊨ T eq . Then every ⊂ na - decomposition inside M extends to a ⊂ na -decomposition of M. Furthermore, if 〈N η , a η : η ∈ I〉 is any ⊂ na -decomposition of M, then M is minimal over ∪N η and for all η ∈ I. M is dominated by ∪N η over N η . Using some stable group theory we show that when Th is superstable with NDOP and 〈 N η : η ∈ I 〉 is a tree decomposition of M, then M is constructible over ∪ n η with respect to a very strong isolation relation. (shrink)

We present a structure theorem for superstable quasi-varieties without DOP. We show that every algebra in such a quasi-variety weakly decomposes as the product of an affine algebra and a combinational algebra, that is, it is bi-interpretable with a two sorted structure where one sort is an affine algebra, the other sort is a combinatorial algebra and the only non-trivial polynomials between the two sorts are certain actions of the affine sort on the combinatorial sort.