Submission date: 5 July 2012.

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 thorn-rank 1 structures and that lovely pairs of weakly one-based geometric thorn-rank 1 struc- tures are weakly one-based with respect to thorn-independence. We also study geometries arising from innite dimensional vector spaces over division rings.

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