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

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    $\begingroup$ Semigroups are much more complicated than this (one problem being that studying congruences on semigroups doesn't reduce to studying certain subsemigroups, and another being that Lagrange's theorem fails miserably). For example, a finitely generated commutative idempotent semigroup is a finite semilattice (en.wikipedia.org/wiki/Semilattice). $\endgroup$ Commented Jun 29, 2013 at 18:12

1 Answer 1


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.


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