Degrees of a primitive element
Definition
Let \(S\) be a numerical semigroup minimally generated by \(P(S) = \{n_1, \ldots, n_e\}\). Let us define \(M \subseteq \mathbb{N}^{2e}\) as the set of all solutions of the equation
\[ P \cdot x - P \cdot y = 0 \Longleftrightarrow (n_1, \ldots, n_e) \cdot x - (n_1, \ldots, n_e) \cdot y = 0, \]
where \(x,y \in \mathbb{N}^e\) and the dot denotes the usual dot product. From the definition, if \((x,y) \in M\), then \(x,y \in \mathbf{Z}(s)\) for a certain \(s \in S\), where \(\mathbf{Z}(s)\) denotes the set of factorizations of \(s\) in \(S\).
It can be proven that \(M\) is a monoid, and then there exists a unique minimal system of generators \(P_M\). It is defined the primitive elements of \(S\) as the elements of \(P_M\), and the degree of a primitive element as the elements \(s \in S\) such that \((x,y) \in \mathbf{Z}(s)\) for some \((x,y) \in P_M\). If \((x,y) \in P_M\) is such that \(|x| \le |y|\), where \(|x|, |y|\) denotes the lengths of \(x\) and \(y\) respectively, it is said that \(s\) is a monotone degree of a primitive element.
It can be proven that Betti elements are primitive.
Examples
\(\circ\) Let \(S\) be a numerical semigroup minimally generated by \(P = \{n_1, \ldots, n_e\}\) and \(M\) the set of solutions of the equation \(p \cdot x - P \cdot y = 0\), minimally generated by \(P_M\). If \(e_i \in \mathbb{N}^e\) is such that the unique coordinate nonzero is \(i-\)th coordinate with value one, then \((e_i, e_i)\) is a primitive element. Indeed, the pair \((e_i, e_i)\) is a solution, since
\[ P \cdot e_i - P \cdot e_i = n_i - n_i = 0, \]
and is minimal for the fact that if \((e_i,e_i) = (x,y) + (x,t)\) for some \((x,y), (z,t) \in M\), then \((x,y) = (0,0)\) or \((z,t) = (0,0)\). This example shows that there are primitive elements that are not minimal relations of \(S\).
\(\circ\) Let \(S = \langle 4, 6, 11 \rangle\). The primitive elements of \(S\) are the minimal solutions of the equation
\[ 4(x_1 - y_1) + 6(x_2 - y_2) + 11(x_3 - y_3) = 0. \]
Applying elimination theorem and extension theorem from ring theory in several indeterminate, it is obtained that
\[ M = \langle (1,0,0,1,0,0), (0,1,0,0,1,0), (0, 0, 2, 1, 3, 0), (0, 0, 2, 4, 1, 0), (0, 0, 4, 11, 0, 0), (0, 0, 6, 0, 11, 0), \\ \]
\[ (0, 1, 2, 7, 0, 0), (0, 2, 0, 3, 0, 0), (0, 5, 0, 2, 0, 2), (0, 8, 0, 1, 0, 4), (0,0,1,0,0,1) \rangle, \]
and the degree of primitive elements of \(S\) is \(\{4, 6, 11, 12, 22, 28, 30, 44, 48, 66\}\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 5, 9, 13, 16 \rangle\), in GAP:
gap> S := NumericalSemigroup(5, 9, 13, 16);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\), the function DegreesOfPrimitiveElementsOfNumericalSemigroup returns the set of degrees of primitive elements of \(S\).
gap> DegreesOfPrimitiveElementsOfNumericalSemigroup(S);
[ 5, 9, 13, 16, 18, 25, 26, 29, 31, 32, 35, 36, 39, 40, 41, 42, 45, 48, 52, 53, 55, 57, 58,
63, 64, 65, 68, 78, 80, 81, 85, 90, 91, 96, 104, 112, 117, 130, 144, 169, 208 ]\(\diamond\) Let \(S = \langle 3, 8, 10 \rangle\), in GAP:
gap> S := NumericalSemigroup(3, 8, 10);
<Numerical semigroup with 3 generators>Given a numerical semigroup \(S\) minimally generated by \(P\), the function DegreesOfEqualPrimitiveElementsOfNumericalSemigroup returns the set of degrees of primitive elements \(s\) in \(S\) so that there exists \((x,y) \in P_M \cap \mathbf{Z}(s)\) with \(|x| = |y|\). Furthermore, the function DegreesOfMonotonePrimitiveElementsOfNumericalSemigroup computes the monotone degrees of primitive elements.
gap> DegreesOfPrimitiveElementsOfNumericalSemigroup(S);
[ 3, 8, 10, 16, 18, 20, 24, 30, 40 ]
gap> DegreesOfEqualPrimitiveElementsOfNumericalSemigroup(S);
[ 3, 8, 10, 56 ]
gap> DegreesOfMonotonePrimitiveElementsOfNumericalSemigroup(S);
[ 3, 8, 10, 16, 18, 20, 24, 30, 40, 56 ]\(\diamond\) Given \(M\) and \(m\) integers, the function NumericalSemigroupsWithMaxPrimitiveAndMultiplicity returns a list of numerical semigroups with maximum \(M\) primitive elements and multiplicity \(m\).
gap> N := NumericalSemigroupsWithMaxPrimitiveAndMultiplicity (20, 7);;
gap> Length(N);
96
gap> S := N[96];
<Numerical semigroup with 3 generators>
gap> D := DegreesOfPrimitiveElementsOfNumericalSemigroup(S);
[ 7, 19, 20, 40, 76, 77, 95, 98, 114, 119, 133, 140, 247, 380 ]
gap> Length(D);
14
gap> Multiplicity(S);
7Moreover, given an integer \(M\), the function NumericalSemigroupsWithMaxPrimitive returns a list with of numerical semigroups with maximum primitive \(M\).
gap> T := NumericalSemigroupsWithMaxPrimitive(10);
[ <Numerical semigroup with 2 generators>, <Numerical semigroup with 3 generators>,
<Numerical semigroup with 3 generators>, <Numerical semigroup with 3 generators>,
<Numerical semigroup with 4 generators>, <Numerical semigroup with 3 generators>,
<Numerical semigroup with 3 generators>, <Numerical semigroup with 4 generators>,
<Numerical semigroup with 4 generators>, <Numerical semigroup with 4 generators>,
<Numerical semigroup with 5 generators>, <Numerical semigroup with 2 generators>,
<Numerical semigroup with 3 generators>, <Numerical semigroup with 3 generators>,
<Numerical semigroup with 4 generators>, <Numerical semigroup with 3 generators>,
<Numerical semigroup with 2 generators> ]
gap> Length(T);
17References
https://gap-packages.github.io/ numericalsgps.