Eliahou slices
Definition
Let \(S\) be a numerical semigroup with multiplicity \(m(S)\) and depth and remainder \(q(S), \rho(S)\) respectively. If
\[ I_j = [jm(S) - \rho(S), (j+1)m(S) - \rho(S)) \cap \mathbb{Z}, ~~~~ \text{for all } ~ j \in \{1, \ldots, q(S)\}, \]
it is defined the Eliahou slices of \(S\) as the sets
\[ S_j = S \cap I_j ~~~~ \text{for all } ~ j \in \{1, \ldots, q(S)\}. \]
From the definition, \(I_{q(S)}\) is the threshold interval of \(S\).
Examples
\(\circ\) Let \(S = \langle 7, 8, 13, 17 \rangle = \{0, 7, 8, 13, 14, 15, 16, 17, 20, \rightarrow\}\). Since the multiplicity of \(S\) is \(m(S) = 7\) and its conductor is \(C(S) = 20\), we have \(q(S) = 3\) and \(\rho(S) = 1\). Then,
\[ I_j = [7j - 1, 7(j+1) - 1) \cap \mathbb{Z}, ~~~~ \text{for all } ~ j \in \{1, 2, 3\}. \]
The Eliahou slices of \(S\) are
\[ S_1 = S \cap I_1 = S \cap [6, 13) = \{7, 8\}, ~~~~ S_2 = S \cap I_2 = S \cap [13, 20) = \{13, 14, 15, 16, 17 \}, \]
\[ S_3 = S \cap I_3 = S \cap [20, 26) = \{20, \ldots, 26\}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 9, 14, 21, 31 \rangle\), in GAP:
gap> S := NumericalSemigroup(9, 14, 21, 31);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\), the function EliahouSlicesOfNumericalSemigroup returns a list with the Eliahou slices of \(S\) for all \(j \in \{1, \ldots, q(S)-1\}\).
gap> EliahouSlicesOfNumericalSemigroup(S);
[ [ 9 ], [ 14, 18 ], [ 21, 23, 27, 28 ],
[ 30, 31, 32, 35, 36, 37 ],
[ 39, 40, 41, 42, 44, 45, 46 ] ]In order to obtain \(S_{q(S)} = S \cap I_{q(S)}\), the following functions computes it.
gap> ThresholdIntervalOfNumericalSemigroup := function(S)
> local c, m;
> if not IsNumericalSemigroup(S) then
> Error("The argument must be a Numerical Semigroup");
> fi;
> c := Conductor(S);
> m := Multiplicity(S);
> return [c..(c + m - 1)];
> end;
function( S ) ... endgap> ThresholdIntervalOfNumericalSemigroup(S);
[ 48 .. 56 ]References
https://gap-packages.github.io/ numericalsgps.