V-monoid
Definition
Let \(\mathcal{V}\) be a Frobenius variety. It is said that a submonoid \(M\) of \(\mathbb{N}\) is a \(\mathcal{V}-\)monoid if it can be expressed as the intersection of elements of \(\mathcal{V}\).
It is deduced from the definition that the intersection of \(\mathcal{V}-\)monoids is a \(\mathcal{V}-\)monoid.
Examples
\(\circ\) Let \(\mathcal{V}\) be the set of all numerical semigroups. It is well known that \(\mathcal{V}\) is a Frobenius variety. Then, the \(\mathcal{V}-\)monoids correspond to the elements of \(\mathcal{V}\) and the monoids generated by no finite intersections of numerical semigroups. For example, if we consider for each \(i \in \mathbb{N} \setminus \{0\}\) the half-line \(S_i = \{0, i, \rightarrow\}\), it holds that
\[ \bigcap_{i = 1}^{+ \infty} S_i = \{0\}, \] is a \(\mathcal{V}-\)monoid, but \(\{0\}\) is not an element of \(\mathcal{V}\).