Unique Betti element numerical semigroup
Definition
Let \(S\) be a numerical semigroup. It is said that \(S\) is a unique Betti element numerical semigroup if \(S\) has a unique Betti element, that is,
\[ |Betti(S)| = |\{s \in S ~ | ~ \nabla_s ~ \text{ is not connected} ~ \}| = 1, \]
where \(\nabla_s\) denotes the graph associated to s by set of factorizations.
It can be proven that if \(S\) has a unique Betti element, then \(S\) is telescopic and \(\alpha-\)rectangular.
Examples
\(\circ\) Let \(S = \langle 10, 18, 45 \rangle\) and \(n_1 = 10 n_2 = 18, n_3 = 45\). It is well known that if \(s \in S\) is a Betti element, then \(s = w + n_i\) with \(w \in Ap(S, n_1) \setminus \{0\}\) and \(i = 2,3\), where \(Ap(S, n_1)\) denotes the Apéry set of \(n_1\) in \(S\). With few calculations,
\[ Ap(S, 10) = \{0, 81, 72, 63, 54, 45, 36, 117, 18, 99\}, \] and the candidates to be Betti elements are
\[ C := \{18, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 162 \}. \]
Computing the graph \(\nabla_s\) for each \(s \in C\), it is concluded that \(\nabla_{90}\) is the unique non-connected graph and \(Betti(S) = \{90\}\). Therefore, \(S\) has unique Betti element.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 26, 59, 61, 68, 75, 80, 86, 95, 96, 116 \rangle\), in GAP:
gap> S := NumericalSemigroup(26, 59, 61, 68, 75, 80, 86, 95, 96, 116);
<Numerical semigroup with 10 generators>Given a numerical semigroup \(S\), the function BettiElements returns a list with the Betti elements of \(S\).
gap> BettiElements(S);
[ 120, 122, 127, 136, 139, 147, 148, 154, 155, 156, 160, 163, 164, 170, 171, 172, 175, 176, 177,
181, 182, 184, 190, 191, 192, 193, 196, 202, 204, 209, 211, 212, 218, 220, 225, 232, 236 ]Therefore, \(S\) has not a unique Betti element. From this function it can be defined a function such that given a numerical semigroup \(S\), it returns true if \(S\) has unique Betti element and false otherwise.
gap> IsUniqueBettiElementNumericalSemigroup := function(S)
> if not IsNumericalSemigroup(S) then
> Error("The argument must be a Numerical Semigroup");
> fi;
> return Length(BettiElements(S)) = 1;
> end;References
https://gap-packages.github.io/ numericalsgps.