R-monoid

Definition

Let \(A\) be a monoid. It is said that \(A\) is a \(\mathcal{R}-\)monoid if it is a cancellative monoid that is not a group and with at least an Archimedean element.

Examples

\(\circ\) \((\mathbb{N}, +)\) is an \(\mathcal{R}-\)monoid. In particular, every numerical semigroup is a \(\mathcal{R}-\)monoid.

\(\circ\) Let \(R\) be an unitary ring and \(A = (M_2(R), \cdot)\), where \(M_2(R)\) is the set of \(2 \times 2\) matrix with coefficients in \(R\) and \(\cdot\) denotes the usual product matrix. Since \(A\) is not cancellative, \(A\) is not a \(\mathcal{R}-\)monoid.

References

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