Weakly one-based geometric theories

Alexander Berenstein (1, 2) , Evgueni Vassiliev (3)

1 ICJ - Institut Camille Jordan
2 UNIANDES - Departamento de Matematicas [Univ de los Andes Colombia]
3 Memorial University - Sir Wilfred Grenfell College - Department of Mathematics

We study the class of weakly locally modular geometric theories introduced in \cite{BeVa}, 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 and the absence of type definable almost quasidesigns. Among other things, we show that weak one-basedness is closed under reducts and generic predicate expansions. We also show that a lovely pair expansion of a non-trivial weakly one-based $\omega$-categorical superrosy thorn rank 1 theory interprets an infinite vector space over a finite field.

Mots clés: Geometric structures; Linear structures; Rosy theories; Geometries; Lovely pairs

Domaines: Logique [math.LO]

Alexander Berenstein, Evgueni Vassiliev. Weakly one-based geometric theories. 2010. ⟨hal-00488140⟩