V-monoid generated by a subset
Definition
The definition of \(\mathcal{V}-\)system of generators comes from the following result.
Let \(\mathcal{V}\) be a Frobenius variety and let \(X\) be a subset of \(\mathbb{N}\). It is defined the \(\mathcal{V}-\)monoid generated by \(X\), denoted by \(\mathcal{V}(X)\), as the intersection of all \(\mathcal{V}-\)monoids that contains \(X\). The subset \(X\) is called a \(\mathcal{V}-\)system of generators of \(\mathcal{V}(X)\) and it is called minimal if no proper subset of \(X\) is a \(\mathcal{V}-\)system of generators.
From the definition, \(\mathcal{V}(X)\) is the smallest \(\mathcal{V}-\)monoid that contains \(X\). It can be proven that every \(\mathcal{V}-\)monoid has a unique minimal \(\mathcal{V}-\)system of generators.
Examples
\(\circ\) Let \(X = \langle 4, 6, 7 \rangle\) and \(\mathcal{V}\) the family of Arf numerical semigroups, which is a Frobenius Variety. Since \(X\) is a numerical semigroup, if \(X \subseteq M\) with \(M\) a \(\mathcal{V}-\)monoid, then \(M\) is a numerical semigroup. Therefore, \(\mathcal{V}(X)\) is the Arf closure of \(X\), which is \(S = \langle 4, 6, 7 \rangle\). It can be proven that \(A = \{4,6,7\}\) is the minimal \(\mathcal{V}-\)system of generators of \(S\) (see Arf system of generators).