TITLE: Structure theorem for finitely generated commutative $semi$groups.
QUESTION [9 upvotes]: $\newcommand{\ZZ}{\mathbb{Z}}$ 
It is a classical theorem that a finitely generated commutative group is isomorphic to one of the form:
$$ \ZZ^n \oplus \ZZ/{m_1}\ZZ \oplus \ZZ/{m_2}\ZZ \oplus \dots \oplus \ZZ/{m_r}\ZZ.$$
In other words, one can construct any finitely generated commutative group out of the "building blocks'' $\ZZ$ and $\ZZ/{m}\ZZ$ for $m = 2,3,4,\dots$. 
I have been wondering: is there a similar structure theorem for semigroups? I am hoping for a result which says something in the spirit of: "There are the following semigroups $(S_i)_{i \in I}$, such that if $S$ is a finitely generated commutative semigroup, then $S \simeq S_{i_1} \oplus S_{i_2} \oplus \dots \oplus S_{i_r}$''.
Surely, for semigroups one needs to allow for more general "building blocks'' than for groups. Also, a simple example of the semigroup $\mathbb{N}_2 := \{2,3,4,\dots\}$ shows that the basic semigroups won't be generated by just a single element. However, I am hoping that there might exist a managable list of semigroups which will suffice.

REPLY [7 votes]: No, the direct product doesn't play the same role for commutative semigroups as for groups. Instead, one uses the so called "semilattices of semigroups". The main theorem (of T. Tamura): Every commutative semigroup is a semilattice of Archimedean semigroups. 
You can find this theorem and necessary definitions in every semigroup book, e.g.:
A. H. Clifford, G. B. Preston, The algebraic theory of semigroups,
J. M. Howie, An introduction to semigroup theory.