Frobenius variety
Definition
Let \(V\) be a non-empty set of numerical semigroups. It is said that \(V\) is a Frobenius variety if fulfill the following conditions:
If \(S, T \in V\), then \(S \cap T \in V\),
If \(S \in V\) and \(S \ne \mathbb{N}\), then \(S \cup \{F(S)\} \in V\),
where \(F(S)\) denotes the Frobenius number of \(S\). It is well known that the intersection of two numerical semigroups is also a numerical semigroup and for any numerical semigroup \(S\), the set \(S \cup \{F(S)\}\) is also a numerical semigroup. Therefore, the conditions of Frobenius variety are well defined.
Examples
\(\circ\) Let \(A\) be a subset of \(\mathbb{N}\), \(\mathcal{S}\) the set of numerical semigroups and let us consider the set \(V = \{S \in \mathcal{S} ~ | ~ A \subseteq S \}\). \(V\) is a Frobenius variety:
If \(S,T \in V\), by definition \(A \subseteq S\) and \(A \subseteq T\), then \(A \subseteq S \cap T\) and \(S \cap T \in V\).
If \(S \in V\), we have \(A \subseteq S \subsetneq S \cup \{F(S)\}\), Therefore, \(S \cup \{F(S)\} \in V\).