Supersymmetric numerical semigroup
Definition
Let \(S\) be a numerical semigroup with Frobenius number \(F(S)\), multiplicity \(m\) and Apéry set \(Ap(S,m)\) in \(m\). It is said that \(S\) is supersymmetric if it is symmetric, additive and whenever \(w + w' = F(S) + m\) for some \(w, w' \in Ap(S,m)\), we have \(ord(w + w') = ord(w) + ord(w')\), where \(ord(\cdot)\) denotes the order of the element.
Examples
\(\circ\) Let \(S = \langle 4, 9, 10 \rangle = \{0, 4, 8, 9, 10, 12, 13, 14, 16, \rightarrow\}\). Since \(F(S) = 15\) and \(g(S) = 8\), where \(g(S)\) denotes the genus of \(S\), we have that \(2g(S) = F(S) + 1\) and \(S\) is symmetric. Moreover, it can be proven that \(adj(S) = Ap(B(S), 4)\), where \(adj(S)\) is the adjustment and \(B(S)\) is the blowup of \(S\), and then \(S\) is additive. On the other hand,
\[ Ap(S, 4) = \{0, 9, 10, 19\}, \]
and \(w + w' = F(S) + m = 19\) if \(\{w,w'\} = \{10, 9\}\). Since \(s = 9\) and \(s = 10\) are minimal generators, we have \(ord(9) = ord(10) = 1\). Finally, \(ord(19) = 2\) and \(S\) is supersymmetric.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 46, 47, 48, 49, 50, 69, 70 \rangle\), in GAP:
gap> S := NumericalSemigroup(46, 47, 48, 49, 50, 69, 70);
<Numerical semigroup with 7 generators>Given a numerical semigroup \(S\), the function IsSuperSymmetricNumericalSemigroup returns true if \(S\) is supersymmetric and false otherwise.
gap> IsSuperSymmetricNumericalSemigroup(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.