On pseudolinearity and generic pairs

摘要

We continue the study of the connection between the "geometric" properties of SU-rank 1 structures and theproperties of "generic" pairs of such structures, started in [8]. In particular, we show that the SU-rank of the(complete) theory of generic pairs of models of an SU-rank 1 theory T can only take values 1 (if and only ifT is trivial), 2 (if and only if T is linear) or w, generalizing the corresponding results for a strongly minimal Tin [3]. We also use pairs to derive the implication from pseudolinearity to linearity for w-categorical SU-rank 1structures, established in [7], from the conjecture that an w-categorical supersimple theory has finite SU-rank,and find a condition on generic pairs, equivalent to pseudolinearity in the general case.