The tensor product of commutative semigroups
Pierre-Antoine Grillet · Transactions of the American Mathematical Society · 1969
In an earlier paper we defined and proved some properties of the tensor product of semigroups.It turns out that the category of commutative semigroups has also a tensor product, which is in many respects more interesting.It keeps all the main properties of the tensor product of arbitrary semigroups (preservation of one-to-one consistent homomorphisms, right exactness, adjoint associativity) and is furthermore colimit-preserving and an associative operation; when applied to abelian groups, it gives the ordinary tensor product of groups.These properties occupy the second section of this paper, the first being devoted to some examples.In the third section, we study flat commutative semigroups.We prove that a flat commutative semigroup with a minimal generating subset must be free, but that flatness is not hereditary and that there exist flat commutative semigroups which are not free.The nature of flat commutative semigroups remains open, as well as the noncommutative case.Heavy use is made of the properties of the tensor product of arbitrary semigroups, which we established in [5].For the fundamentals of semigroup theory, the reader is referred to [1] and [2].Throughout, the largest commutative homomorphic image of a semigroup A will be denoted by C(A) ; the largest idempotent (resp.normal) homomorphic image of A will be denoted by E(A) (resp.N(A)) (normal means: (xy)n = xnyn for all x,y e A and all n [5], [8]).These are covariant functors, in fact coreflexions, of the category of all semigroups onto the full subcategory under consideration; as coreflexions, they preserve colimits and, for instance, the semigroups Horn (A, B) and Horn (C(A), B) are naturally isomorphic whenever B is commutative (if B is commutative, Horn (A, B) is the (commutative) semigroup of all homomorphisms of A into B with pointwise multiplication; for coreflexions, see [6]).1. Definition, existence, examples.If (A,)^, is a family of semigroups, we say that a mapping s of the cartesian product YJieI A¡ into some semigroup C is I-linear if for every i0 e / the mapping ato ~-> s((tf¡)ie/) is a homomorphism of Aio into C whenever all a, are fixed except aio.If all the At's are commutative and if an /-linear mapping t of n¡sí At into some commutative semigroup /is such that any /-linear mapping s of n¡e/ ^i hrto any commutative semigroup C has the form u ° t for some unique homomorphism uofT into C, then we say that the pair it, T) (or the