Ideal admits the pattern
Definition
Let \(S\) be a numerical semigroup, \(I\) a relative ideal of \(S\) and let \(p(x_1, \ldots, x_n) = a_1x_1 + \cdots + a_nx_n\) be a pattern. It is said that \(I\) admits the pattern \(p\) if \(p(s_1, \ldots, s_n) \in S\) for any non-increasing sequence \(s_1, \ldots, s_n\) in \(I\).
If \(J \subseteq I\) is another relative ideal of \(S\), then it can be proven that if \(I\) admits a pattern \(p\), then \(J\) also admits the pattern \(p\).
Examples
\(\circ\) If \(p\) is a pattern and \(S\) is a numerical semigroup admitting the pattern p, then any relative ideal \(I\) admits the pattern \(p\).
\(\circ\) Let \(p(x_1, \ldots, x_n) = a_1x_1 + \cdots + a_n x_n\) be a pattern such that \(gcd(a_1, \ldots, a_n) = 1\) and let \(S\) be a numerical semigroup. It can be proven that for any relative ideal \(I\) of \(S\), it holds that \(p(I) = \{p(s_1, \ldots, s_n) ~ | ~ s_1 \ge s_2 \ge \cdots \ge s_n, ~~ s_i \in I\}\) is a relative ideal of a numerical semigroup.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 4, 7, 10 \rangle\) and \(I = \{33, 50\} + S\), in GAP:
gap> S := NumericalSemigroup(4, 7, 10);
<Numerical semigroup with 3 generators>
gap> I := [33, 50] + S;
<Ideal of numerical semigroup>Given a strongly admissible pattern \(p\) and a proper ideal \(I\), the function ApplyPatternToIdeal returns the list \([d, p(I)/d]\), where \(d\) is the great common divisor of the coefficients of the pattern \(p\) and \(p(I)/d = \{i/d ~ | ~ i \in p(I)\} \subseteq \mathbb{Z}\).
gap> p := [1, 3, -1];
[ 1, 3, -1 ]
gap> L := ApplyPatternToIdeal(p, I);
[ 1, <Ideal of numerical semigroup> ]
gap> pI := L[2];
<Ideal of numerical semigroup>
gap> SmallElements(pI);
[ 99, 103, 106, 107, 110, 111, 113 ]The function SmallElements returns the small elements of the given relative ideal. Then, \(p(I) = \{99, 103, 106, 107, 110, 111, 113 , \rightarrow\}\).
Moreover, given a pattern \(p\) and two relative ideals \(I,J\), the function IsAdmittedPatternByIdeal returns true if \(p(I) \subseteq J\) and false otherwise. If we consider \(J = S\) as ideal, then this function returns true if \(I\) admits the pattern and false otherwise.
gap> IsAdmittedPatternByIdeal(p, I, 0+S);
trueThen, \(I\) admits the pattern \(p\).
\(\diamond\) Let \(S = \langle 18, 19, 26, 28, 30 \rangle\) and \(I = \{11, 38, 62, 70\} + S\), in GAP:
gap> S := NumericalSemigroup(18, 19, 26, 28, 30);
<Numerical semigroup with 5 generators>
gap> I := [11, 38, 62, 70] + S;
<Ideal of numerical semigroup>Given an admissible pattern \(p\) and a positive integer \(C\), the function BoundForConductorOfImageOfPattern returns an upper bound of the smallest element \(K\) in \(p(I)\) such that all integers larger than \(K\) belong to \(p(I)\) (the image of \(I\) by \(p\)), where \(C\) is assumed to be the conductor of a relative ideal \(I\). The conductor can be computed with function Conductor.
gap> C := Conductor(I);
73
gap> p := [3, -1, 4, 5];
[ 3, -1, 4, 5 ]
gap> BoundForConductorOfImageOfPattern(p, C);
823Then, the above result indicates that at least, \(\{823, \rightarrow\} \subseteq p(I)\).
References
https://gap-packages.github.io/ numericalsgps.