Measurability, modules and generic automorphisms

Originally developed [17] as a generalization of dimension and measure on pseudofinite fields, MS-measurability is a relatively new concept in model theory. We look at MS-measurability in structures which already have a well-understood model theory. We also consider other notions of rank within these structures. Firstly, we show that for any module it suffices to put an MS-measuring function on the positive primitive subgroups for the whole structure to be MS-measurable. We give a precise classification of w-stable Abelian groups in terms of MS-measurability. We also show that in this context MS-measurability is compatible with a R[t]-valued measure. Secondly, we look at MS-measurability in the context of strongly minimal structures. We show that MS-measurability is equivalent to uni- modularity. We also consider the definable multiplicity property. We show that it is incomparable to MS-measurability. We give an example of a finite Morley rank structure without the definable multiplicity property for which we conjecture all strongly minimal definable sets have the definable multiplicity property. Finally, we examine structures with a generic automorphism. We show that if T is an w-stable Abelian group then T A is w-stable if and only if T is MS-measurable. We also give a Morley rank two example in which rank in the fixed field behaves unlike rank in the fixed field of a strongly minimal theory with a generic automorphismEThOS - Electronic Theses Online ServiceGBUnited Kingdo