Left elements of a relative ideal
Definition
Let \(S\) be a numerical semigroup and \(E\) a relative ideal of \(S\) with conductor \(C(E)\). It is defined set of left elements of \(E\), denoted by \(N(E)\), as
\[ N(E) = \{e \in E ~ | ~ e < C(E)\}. \] If we also include \(e = C(E)\), then it is called the set of small elements of \(E\).
This definitions is an extension of the definition of left elements of a numerical semigroup, considering \(S\) as ideal of itself.
Examples
\(\circ\) Let \(S = \langle 6, 7, 11 \rangle = \{0, 6, 7, 11, 12, 13, 14, 17, \rightarrow\}\) and \(I = \{-1, 8, 9\} + S\). If \(L = \{0, 6, 7, 11, 12, 13, 14\}\) and \(L - n = \{l - n ~ | ~ l \in L\}\), with few computations,
\[ \begin{align*} I & = (\{-1\} + S) \cup (\{8\} + S) \cup (\{9\} + S) = (L-1) \cup (L+8) \cup (L+9) \cup \{16, \rightarrow\} \\ & = \{ -1, 5, 6, 8, \rightarrow \}. \end{align*} \]
Then, \(C(I) = 8\) and \(N(I) = \{-1, 5, 6\}\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 15, 17, 25, 26, 53, 63 \rangle\) and \(I = \{11, 13, 59\} + S\), in GAP:
gap> S := NumericalSemigroup(15, 17, 25, 26, 53, 63);
<Numerical semigroup with 6 generators>
gap> I := [11, 13, 59] + S;
<Ideal of numerical semigroup>Given a relative ideal \(I\) of a numerical semigroup, the functions SmallElements and SmallElementsOfIdealOfNumericalSemigroup return a list with the small elements of \(I\).
gap> SmallElements(I);
[ 11, 13, 26, 28, 30, 36, 37, 38, 39, 41, 43, 45, 47, 51, 52, 53, 54, 55, 56, 58 ]
gap> SmallElementsOfIdealOfNumericalSemigroup(I) = SmallElements(I);
trueIn function SmallElements, if we give a numerical semigroup \(S\), the function returns the small elements of \(S\).
gap> SmallElements(S);
[ 0, 15, 17, 25, 26, 30, 32, 34, 40, 41, 42, 43, 45, 47, 49, 50, 51, 52, 53, 55, 56, 57,
58, 59, 60, 62 ]References
https://gap-packages.github.io/ numericalsgps.