Minimal set of generators of an ideal
Definition
Let \(S\) be a numerical semigroup and \(E\) a relative ideal of \(S\). It is said that \(I = \{e_1, \ldots, e_h\}\) with \(I \subseteq E\) is a set of generators of \(E\) if \(E = \{e_1, \ldots, e_h\} + S\), where for any two subsets \(A,B \subseteq \mathbb{Z}\), the sum is defined as
\[ A + B = \{a + b ~ | ~ a \in A, ~ b \in B\}. \]Moreover, if no proper subset of \(I\) generates \(E\), then it is said that \(I\) is a minimal set of generators of \(E\).
It can be proven that for every relative ideal \(E\), there exists a unique minimal set of generators, which is \(I = E \setminus (M + E)\), where \(M = S\setminus \{0\}\), and \(I\) is always finite.
Examples
\(\circ\) Let \(S = \langle 4,6,9 \rangle\) and \(E = \{10, 18, 21\} + S\). Considering \(I = \{10, 18, 21\}\), it is clear that \(I\) is a system of generators. Moreover, since \(18 = 10 + 4 \cdot 2 \in \{10, 21\} + S\) and \(21 \not \in \{10\} + S\), it is deduced that \(I^* = \{10, 21\}\) is a minimal set of generators of \(E\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 17, 18, 32, 40, 41, 55 \rangle\) and \(IS = \{-12, -6, 5, 11, 14\} + S\), in GAP:
gap> S := NumericalSemigroup(17, 18, 32, 40, 41, 55);
<Numerical semigroup with 6 generators>
gap> IS := [-12, -6, 5, 11, 14] + S;
<Ideal of numerical semigroup>Given a relative ideal \(IS\), the functions Generators and GeneratorsOfIdealOfNumericalSemigroup return a system of generators of the ideal. The system might not be minimal.
gap> Generators(IS);
[ -12, -6, 5, 11, 14 ]
gap> GeneratorsOfIdealOfNumericalSemigroup(IS);
[ -12, -6, 5, 11, 14 ]On the other hand, the functions MinimalGenerators, MinimalGeneratingSystem and MinimalGeneratingSystemOfIdealOfNumericalSemigroup return a minimal set of generators of the given relative ideal.
gap> MinimalGenerators(IS);
[ -12, -6, 14 ]
gap> MinimalGeneratingSystem(IS);
[ -12, -6, 14 ]
gap> MinimalGeneratingSystemOfIdealOfNumericalSemigroup(IS);
[ -12, -6, 14 ]References
https://gap-packages.github.io/ numericalsgps.