ABSTRACT
If X is any space, the set Vect(X) has the structure of an abelian semigroup, where the additive structure is defined by direct sum. If A is any abelian semigroup, we can associate to A an abelian group K(A) with the following property: there is a semigroup homomorphism α : A → K(A) such that if G is any group, γ : A → G any semigroup homomorphism, there is a unique homomorphism χ : K(A) → G such that γ = χα. If such a K(A) exists, it must be unique.
