Holes
Definition
Let \(S\) be a numerical semigroup with set of gaps \(G(S)\) and Frobenius number \(F(S)\). It is said that \(g \in G(S)\) is a hole of \(S\) if \(F(S) - g \in G(S)\). The set of holes of \(S\), denoted by \(H(S)\), is
\[ H(S) = \{x \in G(S) ~ | ~ F(S) - x \in G(S)\} \subseteq G(S). \]
Since \(F(S) \in G(S)\) and \(F(S) - F(S) = 0 \in S\), it holds that \(H(S) \subsetneq G(S)\) for any numerical semigroup. It can be proven that \(H(S) = \emptyset\) if, and only if, \(S\) is symmetric.
Examples
\(\circ\) Let \(S = \langle 4, 6, 11 \rangle = \{0, 4, 6, 8, 10, 11, 12, 14, \rightarrow\}\). The Frobenius number of \(S\) is \(F(S) = 13\) and the set of gaps is \(G(S) = \{1, 2, 3, 5, 7, 9, 13\}\). If we compute \(F(S) - g\) for all \(g \in G(S)\), we obtain that \(F(S) - G(S) = \{12, 11, 10, 8, 6, 4, 0\} \subseteq S\). Therefore, \(H(S) = \emptyset\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 30, 82, 99, 109, 113 \rangle\), in GAP:
gap> S := NumericalSemigroup(30, 82, 99, 109, 113);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), the functions Holes and HolesOfNumericalSemigroup return the set of holes of \(S\).
gap> Holes(S);
[ 17, 44, 47, 68, 74, 77, 78, 86, 98, 104, 106, 107, 108, 116, 123,
126, 128, 130, 134, 136, 137, 138, 146, 153, 156, 157, 158, 160,
166, 167, 168, 176, 177, 183, 185, 186, 187, 188, 190, 196, 197,
205, 206, 207, 213, 215, 216, 217, 220, 227, 235, 236, 237, 239,
243, 245, 247, 250, 257, 265, 266, 267, 269, 275, 287, 295, 296,
299, 305, 326, 329, 356 ]
gap> Holes(S) = HolesOfNumericalSemigroup(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.