Weak generic types and coverings of groups I

We introduce the notion of a weak generic type in a group.We improve our earlier results on countable coverings of groups and types.Introduction.Assume G is an ℵ 0 -saturated group, or even a group type-denable in an ℵ 0 -saturated structure.Strictly speaking, by a group we sometimes mean an expansion of a pure group structure.Assume G is covered by countably many 0-type-denable sets X n , n < ω.In [NP℄ we proved that in this case some nitely many of the sets X n generate the group G in at most k = 3 steps.More precisely, we proved that (C1) for some nite set A ⊆ G and some n < ω we haveSo in a sense for an arbitrary ℵ 0 -saturated group G, k = 2.5 steps suce to generate G by some nitely many of the sets X n , n < ω.In general, this result cannot be improved (that is, k = 2 steps may not suce).However, in [NP℄ we also proved thatThus in this case k = 2 steps are enough.In this paper we give a new proof of (C1), using the notion of a weak generic type in a group.This new notion generalizes that of a generic type in a stable group, fundamental in stable model theory.Besides its applicability in clarifying (C1) we predict it may play an important role in model theory, particularly in unstable structures (like the o-minimal ones).Regarding (C2), here we extend it in two ways.First, we generalize (C2) to any amenable group G, giving a completely new proof.Secondly, extending 2000 Mathematics Subject Classication: Primary 03C45.