Fundamental gap
Definition
Let \(S\) be a numerical semigroup and let \(G(S)\) be the set of gaps of \(S\). A gap \(g \in G(S)\) is a fundamental gap of \(S\) if \(\{2g, 3g\} \subseteq S\), or equivalently, for every \(n \in \mathbb{N} \setminus \{0,1\}\), it holds that \(ng \in S\). It is denoted by \(FG(S)\) the set of fundamental gaps of \(S\) and is always non-empty, due to \(F(S) \in FG(S)\), where \(F(S)\) is the Frobenius number of \(S\).
It can be proven that \(FG(S)\) determines the numerical semigroup \(S\).
Examples
\(\circ\) Let \(S = \langle 4, 9, 10 \rangle = \{0, 4, 8, 9, 10, 12, 13, 14, 16, \rightarrow \}\). We have that \(G(S) = \{1, 2, 3, 5, 6, 7, 11, 15\}\) and checking the condition for each gap of \(S\), it is concluded that \(FG(S) = \{6, 7, 11, 15\}\).
\(\circ\) Let us consider \(b \in \mathbb{N}\) an odd element arbitrary but fixed. The set \(S = \langle 2, b \rangle\) is a numerical semigroup (note that if \(n \ge b\), then \(n \in S\)) and the gaps of \(S\) are \(G(S) = \{1, 3, 5, 7, \ldots, b - 2\}\), in other words, all the odd naturals smaller than \(b\). Now, a gap \(x \in G(S)\) is fundamental if \(\{2x,3x\} \subseteq S\), in this case, as \(2 \in S\), then \(2x \in S\) for all \(x \in G(S)\). On the other hand, \(3x \in S\) if, and only if, \(x \ge \lceil \frac{b}{3} \rceil\), therefore, \(FG(S)\) consist in all the odd integers greater or equal to \(\lceil \frac{b}{3} \rceil\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 8, 12, 14, 15, 25 \rangle\), in GAP:
gap> S := NumericalSemigroup(8, 12, 14, 15, 25);
<Numerical semigroup with 5 generators>The functions FundamentalGaps and FundamentalGapsOfNumericalSemigroup return the set of fundamental gaps of \(S\).
gap> FundamentalGaps(S);
[ 4, 10, 11, 13, 17, 18, 19, 21 ]
gap> FundamentalGaps(S) = FundamentalGapsOfNumericalSemigroup(S);
trueIf we want to know all the gaps of \(S\), then we have the function Gaps.
gap> Gaps(S);
[ 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 17, 18, 19, 21 ]\(\diamond\) Given a list of positive integers, the function NumericalSemigroupByFundamentalGaps returns the numerical semigroup determined by the list if it is possible, which is considered as the list of fundamental gaps.
gap> l := [6,8,13,15,17,20,22,29];
[ 6, 8, 13, 15, 17, 20, 22, 29 ]
gap> S := NumericalSemigroupByFundamentalGaps(l);
<Numerical semigroup>
gap> MinimalGenerators(S);
[ 7, 9, 12 ]The function MinimalGenerators returns a list with the minimal generators of the numerical semigroup. Then, \(S = \langle 7, 9, 12 \rangle\) and \(FG(S) = \{6, 8, 13, 15, 17, 20, 22, 29\}\).
References
https://gap-packages.github.io/ numericalsgps.