Multiple joined semigroup

Definition

Let \(S\) be a semigroup. It is said that \(S\) is multiple joined if for all \(x, y \in S\), there exist positive integers \(p\) and \(q\) such that \(px = qy\).

Examples

\(\circ\) Given a numerical semigroup \(S\), then \(S^* = S \setminus \{0\}\) is multiple joined.

\(\circ\) Every multiple joined semigroup is Archimedean. Given \(x, y \in S\), there exist \(p, q \in \mathbb{N} \setminus \{0\}\) such that \(px = qy\). Then, \((p+1)x = qy+x\) and \(S\) is Archimedean.

References

Rosales, J. C., and P. A. Garcı́a-Sánchez. 2009. Numerical Semigroups. Springer.