Vol. 32, No. 3, 1970

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Tensor and torsion products of semigroups

Ronald Owen Fulp

Vol. 32 (1970), No. 3, 685–696

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.

Mathematical Subject Classification
Primary: 20.93
Received: 3 November 1967
Revised: 6 September 1969
Published: 1 March 1970
Ronald Owen Fulp