On the wiki page for Grothendieck group, the first sentence says the Grothendieck group is the "best possible" way to construct an abelian group from a commutative monoid.
What actually does this mean formally?
On the wiki page for Grothendieck group, the first sentence says the Grothendieck group is the "best possible" way to construct an abelian group from a commutative monoid.
What actually does this mean formally?
Statements like this usually refer to universality. Note that given a monoid $M$ the Grothendieck group $G$ comes with a map of underlying monoids $s:M\to G$.
Now given $H$ an abelian group and $f:M\rightarrow H$ a map of underlying monoids. Then there exists a unique group homomorphism $g:G\to H$ making the diagram commutative, i.e $g\circ s=f$. This also makes $G$ unique up to isomorphism.
You will find universal properties of that kind at many places in mathematics.
Addendum: The following more or less reflects my personal taste and therefore might or might not be helpful. Whenever I see a universal property I don't fully understand, I convince myself that it makes sense in a simpler setting I can grasp better. So if you want to replace the words monoid by subset of $\mathbb R^n$, homomorphism by inclusion and group by closed set. What is the closed set which approximates a given arbitrary set in the best possibly way? Does this satisfy a similar universal property?