Grothendieck group of an Abelian monoid

Definition
Let $$(M,+,0)$$ be an Abelian monoid. The Grothendieck group of $$M$$ is an Abelian group defined (as a set) as the set of all formal differences $$a - b$$, with $$a,b \in M$$, quotiented by the equivalence relation:

$$a - b = c - d \iff \exists w, a + d + w = b + c + w$$

The group operations are as follows:


 * 1) $$(a - b) + (c - d) := (a + c) - (b + d)$$
 * 2) The identity element is $$0 - 0$$ (which is equal to $$a - a$$ for every $$a$$
 * 3) The inverse of $$a - b$$ is $$b - a$$

Further, there is a natural monoid homomorphism from $$M$$ to $$G$$, given by:

$$a \mapsto a - 0$$

The monoid homomorphism is injective if and only if $$M$$ is a cancellative monoid.