Arf system of generators
Definition
Let \(S\) be an Arf numerical semigroup and \(M = \{m_1, \ldots, m_k\}\) a set of positive integers with \(\gcd(M) = 1\) and \(m_1 < \ldots < m_k\). It is said that \(M\) is an Arf system of generators of \(S\) if the Arf closure of \(M\) is \(S\). If no proper subset of \(M\) has Arf closure \(S\), then it is said that \(M\) is minimal. This definition is a particular case of V-monoids generated by a subset.
It can be proven that for any Arf numerical semigroup \(S\), there exists a finite and unique minimal Arf system of generators \(M\) such that the Arf closure of \(M\) is \(S\). Moreover, \(M\) is a subset of the minimal set of generators of \(S\).
Examples
\(\circ\) Let us consider the Arf semigroup \(S = \langle 4, 6, 7, 9 \rangle = \{0, 3, 6, \rightarrow\}\). From the condition \(gcd(M) = 1\), the candidates are
\[ M_1 = \{4, 7\}, ~~ M_2 = \{4, 9\}, ~~ M_3 = \{6, 7\}, ~~ M_4 = \{7,9\}, \]
\[ M_5 = \{4, 6, 7\}, ~~ M_6 = \{4, 6, 9\}, ~~ M_7 = \{6, 7, 9\}, ~~ M_8 = \{4, 6, 7, 9\}. \]
Computing the Arf closure of each \(M_j\) for \(j \in \{1, \dots, 8\}\), it is obtained that \(M_5 = \{4, 6, 7\}\) is the Arf system of generators of \(S\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 8, 10, 12, 14, 19, 21, 23, 25 \rangle\), in GAP:
gap> S := NumericalSemigroup(8, 10, 12, 14, 19, 21, 23, 25);
<Numerical semigroup with 8 generators>Given a numerical semigroup \(S\), the function IsArf returns true if \(S\) is an Arf numerical semigroup and false otherwise.
gap> IsArf(S);
trueGiven a numerical semigroup \(S\), the functions ArfCharactersOfArfNumericalSemigroup and MinimalArfGeneratingSystemOfArfNumericalSemigroup return the Arf system of generators of \(S\).
gap> ArfCharactersOfArfNumericalSemigroup(S);
[ 8, 10, 19 ]
gap> MinimalArfGeneratingSystemOfArfNumericalSemigroup(S);
[ 8, 10, 19 ]References
https://gap-packages.github.io/ numericalsgps.