Weak one-basedness (Q372621)

In the paper under review, the authors study the notion of weak one-basedness, which was introduced by \textit{A. Berenstein} and \textit{E. Vassiliev} [J. Symb. Log. 77, No. 2, 392--422 (2012; Zbl 1405.03077)]. Variants and related notions such as (very) weak local modularity or linearity (all equivalent in a strongly minimal theory) are compared to weak one-basedness, the outcome being that weak one-basedness characterises linearity in the context of geometric theories of thorn rank 1. This generalises a result of Berenstein and Vassiliev from [loc. cit.].NEWLINENEWLINEFurthermore, it is shown that if \(T\) is a geometric theory of thorn rank 1 which is weakly one-based, then the theory \(T_P\) of lovely pairs of models of \(T\) is also weakly one-based (with respect to thorn-independence).NEWLINENEWLINEIn an appendix, the authors give a proof of the following result (which seems to be folklore). Let \(F\) be a division ring, and let \(V\) be an infinite-dimensional vector space over \(F\). Then the geometry of \(V\) (considered as a first-order structure) is stable if and only if \(F\) is stable.