Suppose is a cancellative abelian semigroup with zero element and we have the homomorphism
where is a abelian group, how do I extend this homomorphism to the whole of where is the enveloping Gothendieck group of to obtain the homomorphism
Elements of are (equivalence classes of) differences of pairs of elements of N. If then define . You need to show that this is well-defined. In other words, you must show that if then . Once you have done this, it is more or less obvious that is a homomorphism.