Are there published results providing characterizations of commutative non-cancellative Archimedean semigroups with no idempotents?

Question

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.)

Answer 1

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 $ac = bc = (1, 2)$.

Answer 2

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.

Answer 3

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.