Profile
Definition
Let \(S\) be a numerical semigroup minimally generated by \(P(S)\) with Eliahou slices \(S_1, \ldots, S_q\) and depth \(q(S)\). If
\[ P_j = P(S) \cap S_j ~~ \text{and} ~~ p_j = |P_j| ~~ \text{for all } ~ j \in \{1, \ldots, q(S)-1\}, \]
it is defined the profile of \(S\) as the \((q(S)-1)-\)tuple \((p_1, \ldots, p_{q(S)-1}) \in \mathbb{N}^{q(S) - 1}\).
Examples
\(\circ\) Let \(S = \langle 4, 9, 10 \rangle = \{ 0, 4, 8, 9, 10, 12, 13, 14, 16, \rightarrow\}\). Since \(m(S) = 4\) and \(C(S) = 16\), where \(m(S)\) and \(C(S)\) denotes the multiplicity and the conductor of \(S\), respectively, we have that \(q(S) = 4\) and remainder \(\rho(S) = 0\). Then,
\[ S_1 = S \cap [4, 8) = \{4\}, ~~ S_2 = S \cap [8, 12) = \{8,9,10\}, ~~ \text{and } ~~ S_3 = S \cap [12, 16) = \{12,13,14\}, \]
concluding that the profile of \(S\) is \((1, 2, 0)\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 19, 31, 60, 78, 80 \rangle\), in GAP:
gap> S := NumericalSemigroup(19, 31, 60, 78, 80);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\), the function ProfileOfNumericalSemigroup returns the profile of \(S\). On the other hand, the function EliahouSlicesOfNumericalSemigroup returns the Eliahou slices \(S_1, \ldots, S_{q(S)-1}\) of the given numerical semigroup.
gap> ProfileOfNumericalSemigroup(S);
[ 2, 0, 1, 2, 0, 0, 0, 0 ]
gap> EliahouSlicesOfNumericalSemigroup(S);
[ [ 19, 31 ], [ 38, 50 ], [ 57, 60, 62, 69 ],
[ 76, 78, 79, 80, 81, 88 ],
[ 91, 93, 95, 97, 98, 99, 100, 107 ],
[ 109, 110, 111, 112, 114, 116, 117, 118, 119, 120,
122, 124, 126 ],
[ 128, 129, 130, 131, 133, 135, 136, 137, 138, 139,
140, 141, 142, 143, 145 ],
[ 147, 148, 149, 150, 151, 152, 153, 154, 155, 156,
157, 158, 159, 160, 161, 162, 164 ] ]References
https://gap-packages.github.io/ numericalsgps.