Set of lengths of an element
Definition
Let \(S\) be a numerical semigroup minimally generated by \(\{n_1, n_2, \ldots, n_p\}\), let \(s \in S\) and let \(\mathbf{Z}(s)\) be the set of factorizations of \(s\). It is defined the set of lengths of \(s\) in \(S\) as
\[ \mathbf{L}(s) = \{|x| ~ | ~ x \in \mathbf{Z}(s)\}, \]
where \(|x|\) denotes the length factorization of \(x\). A factorization \(x \in \mathbf{Z}(s)\) is said to be maximal in \(\mathbf{Z}(s)\) if \(|x|\) is a maximum in \(\mathbf{L}(s)\), that is, \(|x|\) is equal to the order of s in S. The number of maximal factorizations is called the maximal denumerant of \(s\) in \(S\).
Examples
\(\circ\) Let \(S = \langle 5, 12, 17 \rangle\) and \(s = 70\). Its set of factorizations is \(\mathbf{Z}(70) = \{(14, 0), (2,5)\}\). The length of \((14,0)\) is \(14\) and the length of \((2,5)\) is \(7\). Therefore, \(\mathbf{L}(70) = \{14, 7\}\).
\(\circ\) Let \(S = \langle 5, 9, 13 \rangle\) and \(s = 41 \in S\). Its set of factorizations is \(\mathbf{Z}(41) = \{(1,4,0), (2,2,1), (3,0,2)\}\), therefore \(\mathbf{L}(41) = \{5\}\). It is said that a monoid \(A\) is half-factorial if \(|\mathbf{L}(a)| = 1\) for all \(a \in A\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 15, 17, 21, 29 \rangle\), in GAP:
gap> S := NumericalSemigroup(15, 17, 21, 29);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\) and an element \(n\) of it, the function LengthsOfFactorizationsElementWRTNumericalSemigroup computes the set of lengths of \(n\) in \(S\).
gap> LengthsOfFactorizationsElementWRTNumericalSemigroup(120, S);
[ 6, 8 ]Given a list of integers \(Is\) and an element \(n\), the function LengthsOfFactorizationsIntegerWRTList returns the set of lengths of the factorizations of \(n\) in terms of the elements in \(Is\).
gap> LengthsOfFactorizationsIntegerWRTList(120, [10, 20, 35]);
[ 5, 6, 7, 8, 9, 10, 11, 12 ]References
https://gap-packages.github.io/ numericalsgps.