STRONGLY AND CO-STRONGLY MINIMAL ABELIAN STRUCTURES

We give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems: 1. when the only finite 0-definable subgroup is {0}. or equivalently 0 is the only algebraic element (the co-strongly minimal case): 2. when the theory of the structure is strongly minimal. In the first case, we identify the abelian structure as a "near-subspace" A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to ad (empty set)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above. with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d is an element of D. the index of A boolean AND dA in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.