F U N D a M E N T a Mathematicae M-rank and Meager Groups

Assume p∗ is a meager type in a superstable theory T . We investigate definability properties of p∗-closure. We prove that if T has < 2א0 countable models then the multiplicity rank M of every type p is finite. We improve Saffe’s conjecture. 0. Introduction. Throughout the paper, T is a superstable theory in a countable language L, and we work within a monster model C = C of T . The general references are [Ba, Sh, Hru], see also [Ne2]. Suppose p is a regular stationary type. Associated with p is a closure operator clp defined by a ∈ clp(A) iff stp(a) is hereditarily orthogonal to p. Restricted to p(C), clp induces a pre-geometry, and is equivalent to the closure operator induced by forking dependence. For instance, if p is minimal then clp on p(C) equals acl. So when a, b, c are distinct points on a line in p(C) (with respect to the clp-pregeometry), then a ∈ acl(b, c). We show that in fact in many cases clp(a) is definable over clp(b) and clp(c), that is, in the quotient geometry p-closure equals definable closure. These cases include the case of properly weakly minimal p, and more generally of a meager type p. Now let us recall this and other notions introduced in [Ne2, Ne3]. Suppose s(x) is a partial type over C. Then [s] denotes the class of partial types over C, with free variable x, containing s. For any set A let TrA(s) (the trace of s over A) be the set {stp(a/A): a realizes s}. We denote the set of strong types over A by Str(A), and identify it with S(acl(A)). Tr(s) is Tr∅(s). We refer the reader to [Ne3] for the properties of Tr. Sometimes, to specify clearly the variable in the types in question, we write e.g. Strx(A) to denote the set of strong types over A in variable x. We shall often use the following regularity criterion of Hrushovski [Hru]: 1991 Mathematics Subject Classification: Primary 03C45. Research supported by KBN grant 2 P03A 006 09.