All groups considered in this
paper are abelian. It is concerned for the most part with defining suitable tensor
products on categories of partially ordered groups. There is introduced the purely
auxiliary notion of a partial vector space for the purpose of leading to a reasonable
construction of a “vector lattice cover”. The so-called o-tensor product from the
category of p.o. groups into the category of lattice-ordered groups (l-groups) yields
some surprising and surely disappointing results, such as that the functor G ⊗0(.)
preserves monics if and only if G is trivially ordered. This follows from the fact
that if G is trivially ordered then G ⊗0H is independent of the order on H
and in fact l-isomorphic to the free l-group on the ordinary tensor product
G ⊗ H. It should be observed that the latter applies to torsion free groups
only.
