TITLE: Structure Theorem for finitely generated commutative cancellative monoids?
QUESTION [8 upvotes]: 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).

REPLY [4 votes]: Finitely generated commutative semigroups have decidable elementary theories and the isomorphism problem for them is decidable. This means that some sort of description does exist. For example, with every pair of finitely generated commutative monoids, one can associate two tuples of integer matrices such that the monoids are isomorphic iff the tuples are conjugate (Taiclin), and then one can use a result of Grunewald and Segal or Sarkisyan. For references look at the survey http://www.math.vanderbilt.edu/~msapir/ftp/pub/survey/survey.pdf (the part about semigroups there).