Non-existence of universals for classes like reduced torsion free abelian groups under embeddings which are not necessarily pure

Abstract

We consider a class K of structures e.g. trees with omega +1 levels, metric spaces and mainly, classes of Abelian groups like the one mentioned in the title and the class of reduced separable (Abelian) p-groups. We say M in K is universal for K if any member N of K of cardinality not bigger than the cardinality of M can be embedded into M . This is a natural, often raised, problem. We try to draw consequences of cardinal arithmetic to non-existence of universal members for such natural classes.