Arf closure
Definition
Let \(S\) be a numerical semigroup and let \(\mathcal{O}(S)\) be the set of oversemigroups of \(S\). It is defined the Arf closure of \(S\) as the smallest Arf numerical semigroup containing \(S\), and it is denoted by \(Arf(S)\).
It can be proven that the intersection of Arf semigroups is also an Arf semigroup, from which it follows that the Arf closure is the intersection of all Arf numerical semigroups containing it. Since \(\mathcal{O}(S)\) is finite, the intersection is also finite.
Examples
\(\circ\) If \(S\) is an Arf semigroup, then \(Arf(S) = S\).
\(\circ\) Let \(S = \langle 4, 6, 9 \rangle = \{0, 4, 6, 8, 9, 10, 12, \rightarrow \}\). 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 \(S\) is \(A = \bigcap_{i = 0}^8 T_i = \{0, 4, 6, 8, \rightarrow \} = S \cup \{11\}\). In this case, the Arf closure of \(S\) is the numerical semigroup \(S\) to which its Frobenius number has been added.
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 15, 32, 49 \rangle\), in GAP:
gap> S := NumericalSemigroup(15, 32, 49);
<Numerical semigroup with 3 generators>The functions ArfClosure and ArfNumericalSemigroupClosure compute the Arf closure of a numerical semigroup.
gap> A := ArfClosure(S);
<Numerical semigroup>
gap> SmallElements(A);
[ 0, 15, 30, 32, 34, 36, 38, 40, 42, 44 ]
gap> ArfClosure(S) = ArfNumericalSemigroupClosure(S);
trueThe function SmallElements returns a list with the left elements and the conductor of the numerical semigroup. Then, the Arf closure of \(S\) is \(A = \{0,15, 30, 32, 34, 36, 38, 40, 42, 44, \rightarrow \}\).
\(\diamond\) Let \(S = \langle 4, 14, 19, 21 \rangle\), in GAP:
gap> S := NumericalSemigroup(4, 14, 19, 21);
<Numerical semigroup with 4 generators>Given an Arf numerical semigroup \(S\), the function ArfOverSemigroups returns a list of Arf oversemigroups of \(S\).
gap> A := ArfOverSemigroups(S);
[ <The numerical semigroup N>,
<Numerical semigroup with 2 generators>,
<Numerical semigroup with 2 generators>,
<Numerical semigroup with 2 generators>,
<Numerical semigroup with 2 generators>,
<Numerical semigroup with 2 generators>,
<Numerical semigroup with 2 generators>,
<Numerical semigroup with 2 generators>,
<Numerical semigroup with 2 generators>,
<Numerical semigroup with 2 generators>,
<Numerical semigroup with 3 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators> ]
gap> List(A, l -> MinimalGenerators(l));
[ [ 1 ], [ 2, 3 ], [ 2, 5 ], [ 2, 7 ], [ 2, 9 ], [ 2, 11 ],
[ 2, 13 ], [ 2, 15 ], [ 2, 17 ], [ 2, 19 ], [ 3 .. 5 ],
[ 4 .. 7 ], [ 4, 6, 7, 9 ], [ 4, 6, 9, 11 ],
[ 4, 6, 11, 13 ], [ 4, 6, 13, 15 ], [ 4, 6, 15, 17 ],
[ 4, 6, 17, 19 ], [ 4, 6, 19, 21 ], [ 4, 7, 9, 10 ],
[ 4, 9, 10, 11 ], [ 4, 10, 11, 13 ], [ 4, 10, 13, 15 ],
[ 4, 10, 15, 17 ], [ 4, 10, 17, 19 ], [ 4, 10, 19, 21 ],
[ 4, 11, 13, 14 ], [ 4, 13, 14, 15 ], [ 4, 14, 15, 17 ],
[ 4, 14, 17, 19 ], [ 4, 14, 19, 21 ] ]References
https://gap-packages.github.io/ numericalsgps.