A free pseudospace

Abstract

In this paper we construct a non-CM -trivial stable theory in which no infinite field is interpretable. In fact our theory will also be trivial and ω-stable, but of infinite Morley rank. A long term aim would be to find a non CM-trivial theory which has finite Morley rank (or is even strongly minimal) and does not interpret a field. The construction in this paper is direct, and is a “3-dimensional” version of the free pseudoplane. In a sense we are cheating: the original point of the notion of CM-triviality was to describe the geometry of a strongly minimal set, or even of a regular type. In our example, non-CM-triviality will come from the behaviour of three orthogonal regular types.A stable theory is said to be CM-trivial if whenever A ⊆ B and acl(Ac) ∩ acl(B) = acl(A) in Teq, then Cb(stp(c/A)) ⊆ Cb(stp(c/B)). ( An infinite stable field will not be CM-trivial.) The notion is due to Hrushovski [3], where he gave several equivalent definitions, as well as showing that his new strongly minimal sets constructed “ab ovo” were CM-trivial. The notion was studied further in [6] where it was shown that CM-trivial groups of finite Morley rank are nilpotent-by-finite. These results were generalized in various ways to the superstable case in [8].