Is there a Structure Theorem for finitely generated commutative cancellative monoids?

Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, in the book of J. C. Rosales and P. A. García-Sánchez there are some special embedding theorems: If the monoid is torsionfree, it even embeds to some free abelian group, and if the monoid is also reduced, it embeds in some free commutative monoid.

But I want to know if it is possible to give a complete classification (for example, in terms of generators and relations, as in the case of groups).