Telescopic numerical semigroup
Definition
Let \(S\) be a numerical semigroup and \(P = \{n_1, \ldots, n_e\}\) the set of minimal generators of \(S\). It is said that \(S\) is telescopic if \(S\) is free for the arrangement \((n_1, \ldots, n_e)\) and \(n_1 < \cdots < n_e\).
It can be proven that if \(S\) is telescopic, then is \(\beta-\)rectangular.
Examples
\(\circ\) Let \(S = \langle 4, 6, 9 \rangle\). We have, \(d_2 = 4\) and \(d_3 = 2\). Then, we have that \(\overline{c}_2 = \overline{c}_3 = 2\). As \(\overline{c}_2n_2 = 2 \cdot 6 \in \langle 4 \rangle\) and \(\overline{c}_3n_3 = 2 \cdot 9 \in \langle 4, 6 \rangle\), it is concluded that \(S\) is telescopic.
\(\circ\) Let \(S = \langle 4, 9, 14 \rangle\) and the arrangement \(B_1 = \{14, 4, 9\}\). We have, \(d_2 = 14, d_3 = gcd(14, 4) = 2\) and \(d_4 = 1\). Moreover, \(\overline{c}_2 = 7, \overline{c}_3 = 2\), \(\overline{c}_2n_2 = 7 \cdot 4 \in \langle 14 \rangle\) and \(\overline{c}_3n_3 = 2 \cdot 9 \in \langle 14, 4 \rangle\), therefore \(S\) is free for the arrangement \(B_1\). If we check with the arrangement \(B_2 = \{4, 9, 14\}\), is is obtained that \(d_2 = 4, d_3 = gcd(4, 9) = 1, d_4 = 1\) and \(\overline{c}_3 = 1\). Therefore, \(\overline{c}_3 \ne c_3^*\) and \(S\) is not telescopic.
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 4, 11, 14 \rangle\), in GAP:
gap> S := NumericalSemigroup(4, 11, 14);
<Numerical semigroup with 3 generators>In order to know if \(S\) is telescopic, we can use the functions IsTelescopic or IsTelescopicNumericalSemigroup.
gap> IsTelescopic(S);
false
gap> IsTelescopic(S) = IsTelescopicNumericalSemigroup(S);
true\(\diamond\) If we are interested in how many telescopic numerical semigroups have a fixed Frobenius number \(f\), the function TelescopicNumericalSemigroupsWithFrobeniusNumber returns the set of all telescopic numerical semigroups with given Frobenius number \(f\).
gap> Length(TelescopicNumericalSemigroupsWithFrobeniusNumber(11));
4References
https://gap-packages.github.io/ numericalsgps.