We study the class of weakly locally modular geometric theories introduced in , a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over a finite field.
"Weakly one-based geometric theories." J. Symbolic Logic 77 (2) 392 - 422, June 2012. https://doi.org/10.2178/jsl/1333566629