1-based theories and groups
Anand Pillay · 1996
Abstract The notion ‘1-based’ turned up naturally in Chapter 2, as, for example, a way of describing the global behaviour of forking in a superstable theory of finite U-rank all of whose minimal types are locally modular. In this chapter we study arbitrary (stable) theories which are 1-based, and also (∞)-definable sets which are 1-based, without any assumption of finite U-rank, or even superstability. In section 1 we give various equivalences of the 1-based property. In sections 2 and 3 we show that under various additional assumptions (triviality, NDOP, few types) 1-basedness of a theory implies that the theory is ‘superstable-like’ in the sense that types have finite weight, and enough regular types exist. In section 4, we begin to study definable groups in 1-based theories, or even ∞-definable groups which are 1-based. These turn out to be essentially ‘abelian structures’, that is abelian groups equipped with predicates for subgroups. It will also be seen (in Chapter 5) that any ‘non-triviality’ of forking in a stable 1-based theory immediately gives rise to definable groups, so this analysis is quite pertinent. In section 5, we characterize the geometries attached to (locally) modular minimal groups.