Left Elements
Definition
Let \(S\) be a numerical semigroup and \(F(S)\) the Frobenius number of \(S\). An element \(s \in \mathbb{N}\) is a left element of \(S\) if \(s \in S\) and \(s < F(S)\). The set of left elements is denoted by \(N(S)\) or \(L(S)\). If we include the conductor in \(N(S)\), they are said small elements.
The set of left elements is always non-empty, due to \(0 \in N(S)\), and determines the numerical semigroup.
Examples
\(\circ\) Let \(S = \langle 3, 8, 10 \rangle\). We have that \(6 = 3 + 3 \in S\) and since \(8,9,10 \in S\), adding \(3\) to each number, we have that for \(n \ge 8\), it holds \(n \in S\) and \(7 \not \in S\). Then, the Frobenius number is \(F(S) = 7\) and the set of left elements is \(N(S) = \{0, 3, 6\}\).
\(\circ\) Let us consider \(b \in \mathbb{N}\) an odd integer arbitrary but fixed and \(S = \langle 2, b \rangle\). Note that for \(n \ge b-1\), it holds \(n \in S\) and \(b-2 \not \in S\), thus the Frobenius number is \(F(S) = b-2\) and the set of left elements is \(N(S) = \{0, 2, 4, \ldots, b-3\}\), in other words, the even elements between \(0\) and \(b-2\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 4, 13, 17, 20 \rangle\), in GAP:
gap> S := NumericalSemigroup(4, 13, 17, 20);
<Numerical semigroup with 4 generators>If we want to compute the left elements of \(S\), the function SmallElements returns a list with the small elements (left elements and the conductor) of \(S\).
gap> SmallElements(S);
[ 0, 4, 8, 12, 13, 16, 17, 20, 21, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36 ]Obviously, the conductor is the greatest element of the list, in this case, \(C(S) = 36\). Another way to know the left elements of \(S\) is from the function Gaps, which returns the gaps of \(S\). If \(G(S)\) is the set of gaps, therefore \(N(S) = \{0, 1, 2, \ldots, F(S)-1\} \setminus G(S)\).
gap> Gaps(S);
[ 1, 2, 3, 5, 6, 7, 9, 10, 11, 14, 15, 18, 19, 22, 23, 27, 31, 35 ]\(\diamond\) Given a list of non-negative integers, the function NumericalSemigroupBySmallElements generates a numerical semigroup determined by the list if it is possible, which is considered as the list of small elements of the numerical semigroup.
gap> l := [0,5,10,11,14,15,16,19,20,21,22,24];
[ 0, 5, 10, 11, 14, 15, 16, 19, 20, 21, 22, 24 ]
gap> S := NumericalSemigroupBySmallElements(l);
<Numerical semigroup>
gap> MinimalGenerators(S);
[ 5, 11, 14 ]The function MinimalGenerators returns the minimal set of generators of \(S\). It is obtained that \(S = \langle 5, 11, 14 \rangle\) and \(N(S) = \{0, 5, 10, 11, 14, 15, 16, 19, 20, 21, 22\}\).
The function RepresentsSmallElementsOfNumericalSemigroup returns true or false depending on whether a given list may represent the small elements of a numerical semigroup.
gap> RepresentsSmallElementsOfNumericalSemigroup(l);
trueReferences
https://gap-packages.github.io/ numericalsgps.