On pseudolinearity and generic pairs

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 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)