We introduce notions of strong and eventual strong non-isolation for types in countable, stable theories. For T superstable or small stable we prove a dichotomy theorem: a regular type over a finite domain is either eventually strongly non-isolated or is non-orthogonal to a NENI type . As an application we obtain the upper bound for Lascar’s rank of a superstable theory which is one-based or trivial, and has fewer than 20 non-isomorphic countable models.

We prove that a countable, complete, first-order theory with infinite dcl( $ \theta $ ) and precisely three non-isomorphic countable models interprets a variant of Ehrenfeucht’s or Peretyatkin’s example.

Assume T is a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in a type p, in case when p is a complete type over a finite set of parameters. We show that for any flat Morley sequence Q there is a model M of T which is τ-atomic over {Q}. When additionally T has few countable models and is 1-based, we prove that within (...) M there is an infinite Morley sequence I, with I $\subset$ dcl(Q), such that M is prime over I. (shrink)

Assume T is a superstable theory with $ countable models. We prove that any *-algebraic type of M-rank > 0 is m-nonorthogonal to a *-algebraic type of M-rank 1. We study the geometry induced by m-dependence on a *-algebraic type p* of M-rank 1. We prove that after some localization this geometry becomes projective over a division ring F. Associated with p* is a meager type p. We prove that p is determined by p* up to nonorthogonality and that F (...) underlies also the geometry induced by forking dependence on any stationarization of p. Also we study some *-algebraic *-groups of M-rank 1 and prove that any *-algebraic *-group of M-rank 1 is abelian-by-finite. (shrink)

Assume T is stable, small and Φ(x) is a formula of L(T). We study the impact on $T\lceil\Phi$ of naming finitely many elements of a model of T. We consider the cases of $T\lceil\Phi$ which is ω-stable or superstable of finite rank. In these cases we prove that if T has $ countable models and Q = Φ(M) is countable and atomic or saturated, then any good type in S(Q) is τ-stable. If $T\lceil\Phi$ is ω-stable and (bounded, 1-based or of (...) finite rank) with $I(T, \aleph_0) , then we prove that every good p ∈ S(Q) is τ-stable for any countable Q. The proofs of these results lead to several new properties of small stable theories, particularly of types of finite weight in such theories. (shrink)

Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we prove THEOREM A. Let G be a superstable group of U-rank ω such that the generics of G are locally modular and Th(G) has few countable models. Let G - be the group of nongeneric elements of G, G + (...) = G o + G - . Let Π= {q ∈ S( $\emptyset$ ): U(q) $A \subset M$ such that M is almost atomic over A ∪ (G + ∩ M) $\cup \bigcup_{p\in \Pi}$ p(M). (shrink)

We consider a small supersimple theory with a property (CS) (close to stability). We prove that if in such a theoryTthere is a typep∈S(A) (whereAis finite) withSU(p) = 1 and infinitely many extensions overacleq(A), then inTthere is a modular such type. Also, ifTis supersimple with (CS) andp∈S(∅) is isolated,SU(p) = 1 andphas infinitely many extensions overacleq(∅), thenpis modular.

AssumeTis a superstable theory with 0 is m-nonorthogonal to a *-algebraic type of-rank 1. We study the geometry induced by m-dependence on a *-algebraic typep*of-rank 1. We prove that after some localization this geometry becomes projective over a division ring. Associated withp*is a meager typep. We prove thatpis determined byp*up to nonorthogonality and thatunderlies also the geometry induced by forking dependence on any stationarization ofp. Also we study some *-algebraic *-groups of-rank 1 and prove that any *-algebraic *-group of-rank 1 (...) is abelian-by-finite. (shrink)

AssumeTis a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in a typep, in case whenpis a complete type over a finite set of parameters. We show that for any flat Morley sequenceQthere is a modelMofTwhich isτ-atomic over {Q}. When additionallyThas few countable models and is 1-based, we prove that withinMthere is an infinite Morley sequenceI, withI⊂ dcl(Q), such thatMis prime overI.

Assume G is a superstable locally modular group. We describe for any countable model M of Th(G) the quotient group G(M) / Gm(M). Here Gm is the modular part of G. Also, under some additional assumptions we describe G(M) / Gm(M) relative to G⁻(M). We prove Vaught's Conjecture for Th(G) relative to Gm and a finite set provided that ℳ(G) = 1 and the ring of pseudoendomorphisms of G is finite.

Abstract.Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we proveTheorem A.Let G be a superstate group of U-rankωsuch that the generics of G are locally modular andTh(G)has few countable models. Let G−be the group of nongeneric elements of G.G+=Go+G−.LetΠ = {q∈S(∅):U(q)<ω}.For any countable model M ofTh(G)there is a finite (...) A⊂M such thai M is almost atomic over A∪ (G+∩M) ∪ ⋃p∈Πp(M). (shrink)