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The first post below provides examples of commutative non-cancellative Archimedean semigroups with no idempotents. Can anyone provide a reference to a characterization theorem for commutative non-cancellative Archimedean semigroups with no idempotents? (For example it is known that a commutative Archimedean semigroup with an idempotent is an ideal extension of a group by a nilsemigroup.)

Bruce Ebanks
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Let $A$ be the semigroup $(0, 1]$ under the operation $a \cdot b = \min(a + b, 1)$, and let $B$ be the semigroup $\mathbb{N}_+$ under $+$. Let $C = A \times B$ with the product semigroup operation. Both $A$ and $B$ are commutative and Archimedean, so $C$ is as well. Furthermore, $B$ has no idempotents, so $C$ can’t have any idempotents either. However, $C$ is not cancellative. Let $a = (1/2, 1)$ and $b = c = (1, 1)$. Then $a \neq b$, but $ac = bc = (1, 2)$.

Mark Saving
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    Why is $A$ Archimedean? – J.-E. Pin May 15 '23 at 03:23
  • @J.-E.Pin Good point, I actually had a different semigroup in mind than the one I wrote down. The modified definition of $A$ should be correct. In fact, we could even define $A = {0, 1}$ under the operation $x \cdot y = 1$; this should still work. – Mark Saving May 15 '23 at 06:04
  • @MarkSaving Thank you for the examples. Do you know of a result characterizing all such semigroups? I am thinking for example of the result that every commutative Archimedean semigroup with an idempotent is an ideal extension of a group by a nilsemigroup. – Bruce Ebanks May 15 '23 at 19:18
  • @BruceEbanks I have no idea. The first time I considered the matter was when I saw your question. – Mark Saving May 16 '23 at 06:00
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Tamura wrote several paper on this topic, notably

[1] D. Gale and T. Tamura, A theorem on commutative cancellative Archimedean idempotent-free semigroups. (Bulletin of the Belgian Mathematical Society) Simon Stevin 54 (1980), no. 3-4, 233-240.

[2] T. Tamura, Notes on commutative Archimedean semigroups. I, II. Proc. Japan Acad. 42 (1966), 35-40; ibid. 42 (1966) 545-548.

If needed, the complete list of Tamura's articles can be found here.

J.-E. Pin
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Tamura did also study commutative archimedean non-cancellative semigroups. He characterized them by using the notion of structural systems (not very intuitive). Take a look to:

[1] T. Tamura, “Construction of Trees and Commutative Archimedean Semigroups,” Mathematische Nachrichten, vol. 36, no. 5–6. Wiley, pp. 255–287, 1968.

Some open problems about structural system are given at the end of the paper, but I don't know if they were solved or not.

2by2is2mod2
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