On hypergraphs of minimal and prime models of theories of abelian groups

We systematize known results about minimal and prime models of theories of abelian groups. Based on these results we describe properties and provide structural ones on hypergraphs of models for theories of abelian groups. In particular, we characterize non-emptiness and infiniteness of hypergraphs of minimal and prime models. We give necessary and sufficient conditions for almost disjointness of the respective hypergraphs. We also characterize conditions of preservation for non-emptiness and disjointness of hypergraphs of minimal and prime models at transformations to direct sums of groups.