I should read J. C. Rosales and P. A. García-Sánchez's book *Finitely Generated Commutative Monoids* and L. Redei's book *The Theory of Finitely Generated Commutative Semigroups*. I haven't. But here's what I've heard so far:

If a commutative monoid is finitely generated it is finitely presented.

If a finitely generated commutative monoid is

**cancellative**($a + b = a' + b \Rightarrow a = a'$) then it embeds in a finitely generated abelian group.If a finitely generated commutative monoid is cancellative and

**torsion-free**($a + a + \cdots + a = 0 \Rightarrow a = 0$) then it embeds in a finitely generated free abelian group. (This follows easily from the previous claim.)If a commutative monoid is a submonoid of $(\mathbb{N},+,0)$ it is called a numerical monoid and of course it is cancellative. A lot is known about numerical monoids, though I don't believe they have been "classified" in any useful sense.

If we drop the property of being cancellative we get an enormous wilderness of finitely generated commutative monoids, so there shouldn't be any simple 'classification theorem'. But there still might be interesting structure theorems which help us understand this wilderness, just as there are for (say) compact topological abelian groups. What are they?

nowmy last sentence, though it wasn't when Andres Caicedo asked. – John Baez May 20 '13 at 23:53