Generalized almost symmetric numerical semigroup
Definition
Let \(S\) be a numerical semigroup minimally generated by \(P(S)\) with Frobenius number \(F(S)\) and set of pseudo-Frobenius numbers \(PF(S)\), and let \(K(S)\) be its canonical ideal. It is said that \(S\) is a generalized almost symmetric numerical semigroup if either \(2K(S) = K(S)\) (that is, \(S\) is symmetric), where
\[ 2K = \{k + k' ~ | ~ k,k' \in K\}, \]
or \(2K \setminus K = \{F(S) - x_1, \ldots, F(S) - x_r, F(S)\}\) for some \(x_1, \ldots, x_r \not \in P(S)\) such that \(x_i - x_j \not \in PF(S)\).
It can be proven that every almost symmetric numerical semigroup is a generalized almost symmetric numerical semigroup.
Examples
\(\circ\) Let \(S = \langle 10, 11, 19 \rangle\). With few calculations, we have that \(F(S) = 56\) and \(PF(S) = \{47, 56\}\). Since \(F(S)\) is even, \(S\) is not symmetric and \(K(S) \ne K(S)\). It canonical ideal is
\[ K(S) := \{x \in \mathbb{N} ~ | ~ F(S) - x \not \in S \} = \{0,9\} + S, \]
and
\[ 2K \setminus K = (\{0, 9, 18\} + S) \setminus (\{0,9\} + S) = (\{18\} + S) \setminus (\{0,9\} + S) = \{18, 37, 56\}. \] Furthermore, \(x_1 = F(S) - 18 = 38, x_2 = F(S) - 37 = 19\), and it holds \(x_1, x_2 \not \in P(S) = \{10, 11, 19\}\) and \(x_1 - x_2 = 19 \not \in PF(S)\). In conclusion, \(S\) is generalized almost symmetric numerical semigroup.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 11, 14, 15, 16, 17, 18, 20 \rangle\), in GAP:
gap> S := NumericalSemigroup(11, 14, 15, 16, 17, 18, 20);
<Numerical semigroup with 7 generators>Given a numerical semigroup, the function IsGeneralizedAlmostSymmetric returns true if \(S\) is almost symmetric generalized numerical semigroup and false otherwise. On the other hand, the function IsAlmostSymmetric returns true if \(S\) is almost symmetric and false otherwise.
gap> IsGeneralizedAlmostSymmetric(S);
true
gap> IsAlmostSymmetric(S);
falseReferences
https://gap-packages.github.io/ numericalsgps.