Special gap
Definition
Let \(S\) be a numerical semigroup and let \(PF(S)\) be the set of pseudo-Frobenius numbers of \(S\). An element \(x \in PF(S)\) is a special gap if \(2x \in S\). The set of special gaps is denoted by \(SG(S)\) and is always non-empty, due to \(F(S) \in SG(S)\), where \(F(S)\) denotes the Frobenius number of \(S\). Its definition is motivated by the problem of finding the set of all numerical semigroups containing a given numerical semigroup.
Examples
\(\circ\) Let us consider \(m \in \mathbb{N}\) a non-zero element arbitrary but fixed and \(S = \{0, m, \rightarrow \}\). Its easy to see that \(PF(S) = \{1,2, \ldots, m-1\}\), therefore an element \(n \in PF(S)\) is a special gap if \(2n \ge m\), that is, if \(n \ge \lceil \frac{m}{2} \rceil\). In conclusion, \(SG(S) = \{\lceil \frac{m}{2} \rceil, \lceil \frac{m}{2} \rceil + 1, \ldots, m-1 \}\).
\(\circ\) Let \(S = \langle 3, 5, 10 \rangle = \{0, 3, 5, 6, 8, \rightarrow \}\). The positive integers that do not belong to \(S\) are \(\mathbb{N} \setminus S = \{1, 2, 4, 7\}\), and as \(3 \in S\), the candidates for pseudo-Frobenius numbers belongs to the set \(\{-3, -2, -1, 1, 2, 4, 7 \}\). Checking the condition for each element, it is concluded that \(PF(S) = \{7\}\) and \(SG(S) = \{7\}\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 5, 23, 24, 33, 34 \rangle\), in GAP:
gap> S := NumericalSemigroup(5, 23, 24, 33, 34);
<Numerical semigroup with 5 generators>The functions SpecialGaps and SpecialGapsOfNumericalSemigroup returns the set of special gaps of a numerical semigroup.
gap> SpecialGaps(S);
[ 17, 29, 41 ]
gap> SpecialGaps(S) = SpecialGapsOfNumericalSemigroup(S);
true\(\diamond\) Let \(S = \langle 3, 5, 7 \rangle\), in GAP:
gap> S := NumericalSemigroup(3, 5, 7);
<Numerical semigroup with 3 generators>The special gaps of \(S\) are:
gap> SpecialGaps(S);
[ 4 ]For an element to be a special gap, first it has to be a gap and a pseudo-Frobenius number, we can compute this elements with the functions Gaps and PseudoFrobenius, respectively.
gap> Gaps(S);
[ 1, 2, 4 ]
gap> PseudoFrobenius(S);
[ 2, 4 ]With this numerical semigroup we found an example that \(SG(S) \subsetneq PF(S) \subsetneq G(S)\).
References
https://gap-packages.github.io/ numericalsgps.