Pattern
Definition
It is said that an expression \(P\) is a pattern if it is of the form \(a_1x_1 + \cdots + a_nx_n\) with \(x_1, \ldots, x_n\) unknowns and \(a_1, \ldots, a_n\) nonzero integers. If a numerical semigroup \(S\) holds that for every \(s_1, \ldots, s_n \in S\) with \(s_1 \ge s_2 \ge \ldots \ge s_n\), the element \(a_1s_1 + \cdots + a_n s_n\) belongs to \(S\), it is said that \(S\) is a numerical semigroup admitting the pattern. It is denoted by \(\mathcal{S}(P)\) the set of all numerical semigroups admitting the pattern \(P\).
It is said that a pattern \(P\) is admissible if \(\mathcal{S}(P)\) is not empty. If the pattern defined by
\[ P' = \begin{cases} (a_1-1)x_1 + a_2 x_2 + \cdots + a_n x_n & \text{ if } a_1 > 1, \\ a_2 x_2 + \cdots + a_n x_n & \text{ otherwise, } \end{cases} \] is also admissible, it is said that \(P\) is strongly admissible.
Examples
\(\circ\) Let \(P \equiv x_1 + x_2 - x_3\). It holds \(\mathbb{N} \in \mathcal{S}(P)\), since for all \(m_1 \ge m_2 \ge m_3\),
\[ m_1 + m_2 - m_3 = m_1 + (m_2 - m_3) \ge 0. \]
Hence, \(P\) is admissible. On the other hand, \(P' = x_2 - x_3\), which is also admissible. Therefore, \(P\) is strongly admissible. The set \(\mathcal{S}(x_1 + x_2 - x_3)\) is the set of Arf semigroups.
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Given a list of non-negative integers, the function IsAdmissiblePattern returns true or false depending on whether the pattern \(P\) defined by the list is admissible or not.
gap> IsAdmissiblePattern([6, -4, -2]);
trueThat means, there exists a numerical semigroup \(S\) such that admits the pattern \(P \equiv 6x - 4x - 2x\).
\(\diamond\) Given a list of non-negative integers, the function IsStronglyAdmissiblePattern returns true or false depending on whether the pattern \(P\) defined by the list is strongly admissible or not.
gap> IsAdmissiblePattern([7, -2, -5]);
true
gap> IsStronglyAdmissiblePattern([7, -2, -5]);
falseThat means, the set \(\mathcal{S}(P)\) is non-empty but the set \(\mathcal{S}(P')\) it is.
References
https://gap-packages.github.io/ numericalsgps.