Arf special gap
Definition
Let \(S\) be an Arf numerical semigroup. It is defined the set of Arf special gaps of \(S\), denoted by \(ArfG(S)\), as
\[ ArfG(S) = \{x \in SG(S) ~ | ~ S \cup \{x\} ~~ \text{is an Arf numerical semigroup}\}, \]
where \(SG(S)\) denotes the set of special gaps of \(S\).
If \(S = \{h_0 = 0, h_1, \ldots, h_{n-1}, h_n, \rightarrow\}\) is an Arf numerical semigroup and \(x \in G(S)\) is such that \(h_i < x < h_{i+1}\) for some \(i \in \{0, \ldots, n-1\}\), it can be proven that \(x \in ArfG(S)\) if, and only if, \(x \in SG(S)\), \(2x - h_i \in S\) and \(2h_{i+1} - x \in S\).
Examples
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle = \{0, 3, 5, \rightarrow\}\). The set of gaps of \(S\) is \(G(S) = \{1, 2, 4\}\) and its set of special gaps is \(SG(S) := \{x \in G(S) ~ | ~ \{2x, 3x\} \subseteq S\} = \{4\}\). We have that \(3 < 4 < 5\), \(2 \cdot 4 - 3 = 5 \in S\) and \(2 \cdot 5 - 4 = 6 \in S\), concluding that \(ArfSG(S) = \{4\}\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle P(S) \rangle\) with
\[ P(S) = \{26, 29, 32, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 56, 57, 59, 60, 62\}, \]
in GAP:
gap> P := [26, 29, 32, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 56, 57, 59, 60, 62];
[ 26, 29, 32, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
53, 54, 56, 57, 59, 60, 62 ]
gap> S := NumericalSemigroup(P);
<Numerical semigroup with 26 generators>Given a numerical semigroup \(S\), the function ArfSpecialGaps returns the set of Arf special gaps of \(S\).
gap> ArfSpecialGaps(S);
[ 13, 20, 23, 36 ]References
https://gap-packages.github.io/ numericalsgps.