Hereditarily finitely generated monoid

Definition

Let \(A\) be a monoid. It is said that \(A\) is a hereditarily finitely generated monoid if every submonoid is finitely generated.

Examples

\(\circ\) Let us consider \(A = (\mathbb{N}, +)\) and \(X\) a submonoid of \(A\). Let \(d = gcd(X)\) and \(S = \left \{\frac{x}{d} ~ | ~ x \in X \right\}\). Since \(gcd(S) = 1\), it is well known that \(S\) is a numerical semigroup and it has a minimal set of generators \(P\). Then, \(dP = \{dp ~ | ~ p \in P\}\) is a set of generators of \(X\) and \(A\) is a hereditarily finitely generated monoid.

References

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