States Universal Property of the Grothendieck Group of a Commutative Monoid: Let (M,+,0_M) be a commutative monoid, written additively. Then there exist an abelian group G(M) and an identity-preserving monoid homomorphism :M G(M) such that for every abelian group A and every identity-preserving monoid homomorphism f:M A, where A is reg