This paper is concerned with
the study of tensor and torsion functors on the category of abelian semigroups. We
show that such functors exist, that they satisfy the universal diagram properties
required of them in other branches of algebra, and that many of the theorems
obtained for tensor and torsion products of modules may also be obtained in this
setting. In particular the tensor functor ⊗0 is exact relative to the category of
identity preserving homomorphisms. We determine certain structural characteristics
of ⊗. If E and F are maximal semilattice homomorphic images of abelian
semigroups S and T respectively, then E ⊗ F is the maximal semilattice
homomorphic image of S ⊗T. If G and H are maximal subgroups of lS and T then
G ⊗ H may be identified as a subgroup of S ⊗ T and if G and H are the
groups of units of S and T respectively, then G ⊗ H is the group of units
of S ⊗ T. Moreover, the tensor product of abelian inverse semigroups is
an abelian inverse semigroup. Similar results are obtained for the torsion
functor.
