Generic pairs of SU-rank 1 structures

✍ Scribed by Evgueni Vassiliev

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 422 KB
• Volume: 120
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(02)00060-x

✦ Synopsis

For a supersimple SU-rank 1 theory T we introduce the notion of a generic elementary pair of models of T (generic T -pair). We show that the theory T * of all generic T -pairs is complete and supersimple. In the strongly minimal case, T * coincides with the theory of inÿnite dimensional pairs, which was used in (S. Buechler, Pseudoprojective strongly minimal sets are locally projective, J. Symbolic Logic 56(4) (1991) 1184 -1194) to study the geometric properties of T . In our SU-rank 1 setting, we use T * for the same purpose. In particular, we obtain a characterization of linearity for SU-rank 1 structures by giving several equivalent conditions on T * , ÿnd a "weak" version of local modularity which is equivalent to linearity, show that linearity coincides with 1-basedness, and use the generic pairs to "recover" projective geometries over division rings from non-trivial linear SU-rank 1 structures.