Multiplicity of an ideal
Definition
Let \(S\) be a numerical semigroup and let \(E\) be a relative ideal of \(S\). It is defined the multiplicity of \(E\) as the minimum of \(E\).
It can be proven that every relative ideal of a semigroup has a minimum. This property is due to the second condition of relative ideal of a numerical semigroup.
Examples
\(\circ\) Let \(E = \{-2, -1\} + S\). Clearly, \(S + E \subseteq S\) and if we take \(s = C(S) + 2\), where \(C(S)\) is the conductor of \(S\) then \(s + E \subseteq S\). Therefore, \(E\) is a relative ideal of \(S\) and \(m(E) = -2\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(I = \{8, 9, \ldots, 15\}\) and \(S = \langle 6, 10, 15 \rangle\), in GAP:
gap> I := [8..15];
[ 8 .. 15 ]
gap> S := NumericalSemigroup(6,8,15);
<Numerical semigroup with 3 generators>In order to obtain the multiplicity of the relative ideal \(E = I + S\), we can use the function Minimum.
gap> Minimum(I+S);
8References
https://gap-packages.github.io/ numericalsgps.