$\newcommand{\ZZ}{\mathbb{Z}}$ 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.