On pseudolinearity and generic pairs

Abstract

We continue the study of the connection between the "geometric" properties of SU -rank 1 structures and the properties 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 if T is trivial), 2 (if and only if T is linear) or ω, generalizing the corresponding results for a strongly minimal T in [3]. We also use pairs to derive the implication from pseudolinearity to linearity for ω-categorical SU -rank 1 structures, established in [7], from the conjecture that an ω-categorical supersimple theory has finite SU -rank, and find a condition on generic pairs, equivalent to pseudolinearity in the general case.