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

by Shelah. [Sh:552]

Advances in Algebra and Model Theory. Editors: Manfred Droste and Ruediger Goebel, 1997

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.

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