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

Assume T is superstable and small. Using the multiplicity rankM we find locally modular types in the same manner as U -rank considerations yield regular types. We define local versions of M-rank, which also yield meager types. 0. Introduction. Throughout, T is superstable, small, and we work in C = C. In [Ne1] we defined the multiplicity rank M and proved that M has additivity properties similar to those of U -rank. In [Ne2] we defined the notion of meager regular type and proved that every such type is locally modular. It turns out that using M-rank we can produce locally modular regular types of prescribedM-rank. These types are either trivial or meager (the second case always holds when T has < 2א0 countable models). We use the following notation. If s(x) is a (partial) type over C, then [s] denotes the class of (partial) types over C containing s. For p ∈ S(A), St(p) is the set of stationarizations of p over C, StA(p) = {r|acl(A) : r ∈ St(p)}. For B ⊇ A, Sp(B) = S(B) ∩ [p] and Sp,nf(B) = {q ∈ S(B) ∩ [p] : q does not fork over A}. We regard strong types over A as types over acl(A). We define TrA(s) (the trace of s on A) as the set of types r(x) ∈ S(acl(A)) consistent with s(x). Thus if p ∈ S(A) then TrA(p) = StA(p). In general, TrA(s) is closed. TrA(a/B) abbreviates TrA(tp(a/B)). If a | ∪ B (A) then sometimes we use StA(a/B) to denote TrA(a/B). Also, xA denotes the tuple of variables x indexed by elements of A. We will often tacitly use the following easy fact. Fact 0.1. Assume A ⊆ B ⊆ C. Then either TrA(a/C) is open in TrA(a/B) or TrA(a/C) is nowhere dense in TrA(a/B). P r o o f. Suppose TrA(a/C) is not nowhere dense in TrA(a/B). Since TrA(a/C) is closed, this means that for some φ = φ(x, c′) with c′ ∈ acl(A), ∅ 6= [φ] ∩ TrA(a/B) ⊆ TrA(a/C). Thus we can choose a′ realizing φ(x, c′) 1991 Mathematics Subject Classification: Primary 03C45. Research supported by KBN grant 2-1103-91-01.