The tensor product of semigroups

Abstract

The tensor product of semigroups is defined like the tensor product of modules, by means of multilinear mappings. Surprisingly enough, it has most of the important properties of its homological cousin, together with some others of its own, so that, in view of such results as [4], it is not unreasonable to hope that it will become a very helpful tool for the study of semigroups. The only thing it lacks as an operation is associativity; this is apparently due to the use of noncommutative semigroups throughout this paper, and results in our concentrating on the tensor product of two semigroups, even though such restriction is not necessary for some of our results. In ?1 we give the definition and some examples: If B is a one-element semigroup, then A X B is the largest idempotent homomorphic image of A; if B is an infinite cyclic semigroup, then A 0 B is the largest normal homomorphic image N(A) of A (normal means that (Xy)n=Xnyn holds identically for all n). In ?2 we prove the existence of the tensor product of any family of semigroups. ?3 brings a fundamental result, which describes the congruence induced by f X g when the homomorphismsf and g are onto (in which casef 0 g is also onto). As a first consequence, we prove also that A X B depends only on the largest normal homomorphic images of A and B; namely, A 0 B is naturally isomorphic to N(A) X N(B). In ?4, we prove a very peculiar property of the tensor product of semigroups, namely that it preserves consistent monomorphisms (a semigroup homomorphism is consistent if the complement of its image is an ideal or is empty). The fundamental result of ?3 can then be extended to consistent morphisms. In ?5 we show that the tensor product of semigroups is cokernel preserving in the following sense. Call a sequence A' 4 A XL- A' coexact if the congruence induced by f' is the smallest in a class in which the image of f is contained. Then the tensor product by any s-indecomposable semigroup X preserves such coexact sequences where f' is onto (or consistent). We also establish the adjoint associativity. In the right exactness result of ?5, the assumption on X cannot be lifted, but we show in ?6 that this can be fixed by defining another tensor product for semigroups with zero, which is closely related to the tensor product of semigroups and otherwise keeps most of its properties.