Weak one-basedness

Abstract

We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterises linearity in the setting of geometric þ-rank 1 structures and that lovely pairs of weakly one-based geometric þ-rank 1 struc- tures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite dimensional vector spaces over division rings.