Quasi-Archimedean monoid

Definition

Let \(A\) be a monoid with identity element \(0\). It is said that \(A\) is quasi-Archimedean if \(0\) is not an Archimedean \(A \setminus \{0\}\) is Archimedean.

Examples

\(\circ\) Given a numerical semigroup \(S\), it holds that \(S\) is Quasi-Archimedean. First, \(s = 0\) is not an Archimedean. On the other hand, let \(F(S)\) be the Frobenius number of \(S\). Given \(x, y \in S^*\), there exists \(k \in \mathbb{N} \setminus \{0\}\) such that \(kx - y > F(S)\). Then, \(z := kx - y \in S^*\) and \(kx = y + z\).

References

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