Arf closure of a set
Definition
Let \(X\) be a non-empty set of \(\mathbb{N}\) with \(gcd(X) = 1\). It is well known that the condition \(gcd(X) = 1\) is sufficient for \(\langle X \rangle\) to be a numerical semigroup. Moreover, every Arf semigroup that contains \(X\) must contain \(\langle X \rangle\). Then, it is defined the Arf closure of \(X\) as the Arf closure of \(S = \langle X \rangle\).
Examples
\(\circ\) Let \(X = \{4, 6, 9\}\) and \(S = \langle X \rangle = \langle 4, 6, 9 \rangle\). The set of oversemigroups is
\[ \mathcal{O}(S) = \{\mathbb{N}, \{0, 2, \rightarrow \}, \{0, 2, 4, \rightarrow \}, \{0, 2, 4, 6, \rightarrow\}, \{0, 2, 4, 6, 8, \rightarrow \}, \{0, 3, \rightarrow \}, \]
\[ \{0, 3, 4, 6, \rightarrow \}, \{0, 4, \rightarrow \}, \{0, 4, 5, 6, 8, \rightarrow \}, \{0, 4, 6, \rightarrow \}, \{0, 4, 6, 8, \rightarrow \}, S \}. \]
Moreover, the Arf semigroups are
\[ T_0 = \mathbb{N}, T_1 = \{0, 2, \rightarrow \}, T_2 = \{0, 2, 4, \rightarrow \}, T_3 = \{0, 2, 4, 6, \rightarrow \}, T_4 = \{0, 2, 4, 6, 8, \rightarrow \}, \]
\[ T_5 = \{0, 3, \rightarrow \}, T_6 = \{0, 4, \rightarrow \}, T_7 = \{0, 4, 6, \rightarrow \}, T_8 = \{0, 4, 6, 8, \rightarrow \}. \]
Therefore, the Arf closure of \(X\) is \(A = \bigcap_{i = 0}^8 T_i = \{0, 4, 6, 8, \rightarrow \} = S \cup \{11\}\). In this case, the Arf closure of \(X\) is the numerical semigroup \(\langle X \rangle\) to which its Frobenius number has been added.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(X = \{4, 13, 18, 23\}\) and \(S = \langle X \rangle\), in GAP:
gap> x := [4, 13, 18, 23];
[ 4, 13, 18, 23 ]
gap> S := NumericalSemigroup(x);
<Numerical semigroup with 4 generators>The functions ArfClosure and ArfNumericalSemigroupClosure compute the Arf closure of a numerical semigroup.
gap> A := ArfClosure(S);
<Numerical semigroup>
gap> ArfClosure(S) = ArfNumericalSemigroupClosure(S);
trueWe can use the function SmallElements to compute the left elements and the conductor of a numerical semigroup.
gap> SmallElements(A)
[ 0, 4, 8, 12 ]Therefore, the Arf closure of \(X\) is \(A = \{0, 4, 8, 12, \rightarrow\}\).
References
https://gap-packages.github.io/ numericalsgps.