Denumerant ideal
Definition
Let \(S\) be a numerical semigroup and \(n\) a non-negative integer. It is defined the denumerant ideal of \(S\) with respect to \(n\) as the subset of \(S\) defined as
\[ D(S, n) = \{x \in S ~ | ~ |\mathbf{Z}(x)| > n \}, \]
where \(\mathbf{Z}(x)\) denotes the set of factorizations of \(x\) in \(S\). It can be proven that \(D(S, n)\) is always a proper ideal of \(S\). If we add zero to this set, it is called \(n-\)semigroup.
From the definition, if \(n \le m\), then \(D(S, m) \subseteq D(S, n)\).
Examples
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle\) and \(n = 4\). The conductor of \(S\) is \(C(S) = 14\) and considering \(s = 42\), we have that
\[ \mathbf{Z}(42) = \{(7, 1, 0), (0, 6, 0), (1, 4, 1), (2, 2, 2), (3, 0, 3) \} \Longrightarrow 42 \in D(S, 4). \] It is deduced that \(s \in D(S, 4)\) for all \(s \ge C(S) + 42 = 56\) (since \(s - 42 \in S\) and \(42\) has more than \(4\) factorizations). Checking the factorizations on the remaining elements, it is concluded that \(D(S, 4) = \{40, 42, \rightarrow \}\)
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 17, 31, 32, 40, 58 \rangle\), in GAP:
gap> S := NumericalSemigroup(17, 31, 32, 40, 58);
<Numerical semigroup with 5 generators>Given a numerical semigroup \(S\) and a non-negative element \(n \in S\), the function DenumerantIdeal returns the denumerant ideal of \(S\) with respect to to \(n\).
gap> I := DenumerantIdeal(S, 20);
<Ideal of numerical semigroup>
gap> DenumerantIdeal(S, 20) = I;
true
gap> SmallElements(I);
[ 280, 284, 288, 289, 290, 292 ]Therefore, \(D(S, 20) = \{280, 284, 288, 289, 290, 292, \rightarrow\}\).
References
https://gap-packages.github.io/ numericalsgps.