Abstract

We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion ofseparation of quantifierswhich is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one functionμfrom ‘primitive extensions’ to the natural numbers a theoryTμof an expansion of an algebraically closed field which has Morley rank 2. Finally, we show that ifμis not finite-to-one the theory may not beω-stable.