L-shape of Apéry set

Definition

Let \(S\) be a numerical semigroup minimally generated by \(P(S) = \{n_1, \ldots, n_e\}\) and \(m \in S \setminus \{0\}\). It is said that \(L \subseteq \mathbb{N}^e\) is an \(L-\)shape associated to \(Ap(S,m)\) if the following two properties hold.

• The map \((x_1, \ldots, x_e) \to n_1x_1 + \cdots + n_e x_e\) is a bijection from \(L\) to \(Ap(S,m)\).

• If \((x_1, \ldots, x_e) \in L\), then \((y_1, \ldots, y_e) \in L\) for every \((y_1, \ldots, y_e) \in \mathbb{N}^e\) such that \(y_i \le x_i\) for all \(i \in \{1, \ldots, e\}\).

If \(\mathbf{Z}(s)\) is the set of factorizations for any \(s \in S\), the first condition means that \(|\mathbf{Z}(s) \cap L| = 1\) for all \(s \in Ap(s,m)\). Moreover, it can be proven that if \(x \in Ap(S,m)\) and \(x - y \in S\), then \(y \in Ap(S,m)\), and the second condition means that \((y_1, \ldots, y_e)\) is a factorization of an element in \(Ap(S,m)\).

Examples

\(\circ\) Let \(S = \langle 3, 5, 7 \rangle = \{0, 3, 5, \rightarrow \}\). Let us prove that

\[ L = \{ (0, 0, 0), (1, 0, 0), (0, 1, 0), (2, 0, 0), (1, 1, 0), (3, 0, 0), (2, 1, 0) \}, \]

is an \(L-\)shape of \(Ap(S,7)\). From the definitions,

\[ Ap(S, 7) := \{s \in S ~ | ~ s - 4 \not \in S\} = \{0, 8, 9, 3, 11, 5, 6 \}, \]

and

\[ \mathbf{Z}(0) = \{(0,0,0)\}, ~ \mathbf{Z}(8) = \{(1,1,0)\}, ~ \mathbf{Z}(9) = \{(3,0,0)\}, ~ \mathbf{Z}(3) = \{(1,0,0)\}, \]

\[ \mathbf{Z}(11) = \{(2,1,0)\}, ~ \mathbf{Z}(5) = \{(0,1,0)\}, ~ \mathbf{Z}(6) = \{(2,0,0)\}. \]

Since each element has an unique factorization and \(L\) is the union of all factorizations, it is concluded that \(L\) is an \(L-\)shape of \(Ap(S,7)\). In fact, is the unique \(L-\)shape of \(Ap(S,7)\).

Examples with GAP

The following example is made with the package NumericalSgps in GAP.

\(\diamond\) Let \(S = \langle 13, 14, 33, 64 \rangle\), in GAP:

gap> S := NumericalSemigroup(13, 14, 33, 64);
<Numerical semigroup with 4 generators>

Given a numerical semigroup \(S\) minimally generated by \(P = \{n_1 < n_2 < \cdots < n_e\}\), the functions LShapes and LShapesOfNumericalSemigroup return a list with all L-shapes of \(Ap(S, n_e)\).

gap> LShapes(S);
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gap> LShapesOfNumericalSemigroup(S) = LShapes(S);
true

References

Aguiló-Gost, F., P. A. García-Sánchez, and D. Llena. 2015. “On the Number of l-Shapes in Embedding Dimension Four Numerical Semigroups.” Discrete Mathematics 338 (12): 2168–78. https://doi.org/10.1016/j.disc.2015.05.019.

Delgado, M., P. A. Garcia-Sanchez, and J. Morais. 2024. “NumericalSgps, a Package for Numerical Semigroups, Version 1.4.0.” https://gap-packages.github.io/ numericalsgps.