Order of an element

Definition

Let \(S\) be a numerical semigroup and \(s \in S\) with set of lengths \(\mathbf{L}(s)\). It is defined the order of \(s\) in \(S\), denoted by \(ord(s;S)\) or simply \(ord(s)\), as

\[ ord(s) = \max \mathbf{L}(s). \]

That means, the order of \(s\) in \(S\) is the maximum length between its set of factorizations.

Examples

\(\circ\) Let \(S = \langle 10, 12, 15 \rangle\) and \(s = 70\). If \((\alpha, \beta, \gamma) \in \mathbf{Z}(70)\), where \(\mathbf{Z}(70)\) denotes the set of factorizations of \(70\) in \(S\),

\[ 70 = 10 \cdot \alpha + 12 \cdot \beta + 15 \cdot \gamma, \]

obtaining that \(0 \le \alpha \le 7, 0 \le \beta \le 4\) and \(0 \le \gamma \le 5\). From the previous argument, it is deduced that

\[ \mathbf{Z}(70) = \{(7, 0, 0), (1, 5, 0), (4, 0, 2), (1, 0, 4) \}, \]

and \(\mathbf{L}(70) = \{5, 6, 7\}\), concluding that \(ord(70) = 7\).

Examples with GAP

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

\(\diamond\) Let \(S = \langle 12, 27, 28, 70, 101 \rangle\), in GAP:

gap> S := NumericalSemigroup(12, 27, 28, 70, 101);
<Numerical semigroup with 5 generators>

Given a numerical semigroup \(S\) and an element \(n \in S\), the functions MaximumDegree and MaximumDegreeOfElementWRTNumericalSemigroup return the order of \(n\) in \(S\).

gap> 415 in S;
true
gap> MaximumDegree(S, 415);
32
gap> MaximumDegreeOfElementWRTNumericalSemigroup(415, S);
32

On the other hand, given a numerical semigroup \(S\) and an element \(n \in S\), the function Factorizations returns its set of factorizations in \(S\) and LengthsOfFactorizationsElementWRTNumericalSemigroup returns the set of lengths in \(S\).

gap> Factorizations(415, S);
[ [ 30, 1, 1, 0, 0 ], [ 21, 5, 1, 0, 0 ], [ 12, 9, 1, 0, 0 ],
  [ 3, 13, 1, 0, 0 ], [ 23, 1, 4, 0, 0 ], [ 14, 5, 4, 0, 0 ],
  [ 5, 9, 4, 0, 0 ], [ 16, 1, 7, 0, 0 ], [ 7, 5, 7, 0, 0 ],
  [ 9, 1, 10, 0, 0 ], [ 0, 5, 10, 0, 0 ], [ 2, 1, 13, 0, 0 ],
  [ 22, 3, 0, 1, 0 ], [ 13, 7, 0, 1, 0 ], [ 4, 11, 0, 1, 0 ],
  [ 15, 3, 3, 1, 0 ], [ 6, 7, 3, 1, 0 ], [ 8, 3, 6, 1, 0 ],
  [ 1, 3, 9, 1, 0 ], [ 16, 1, 2, 2, 0 ], [ 7, 5, 2, 2, 0 ],
  [ 9, 1, 5, 2, 0 ], [ 0, 5, 5, 2, 0 ], [ 2, 1, 8, 2, 0 ],
  [ 8, 3, 1, 3, 0 ], [ 1, 3, 4, 3, 0 ], [ 9, 1, 0, 4, 0 ],
  [ 0, 5, 0, 4, 0 ], [ 2, 1, 3, 4, 0 ], [ 17, 2, 2, 0, 1 ],
  [ 8, 6, 2, 0, 1 ], [ 10, 2, 5, 0, 1 ], [ 1, 6, 5, 0, 1 ],
  [ 3, 2, 8, 0, 1 ], [ 18, 0, 1, 1, 1 ], [ 9, 4, 1, 1, 1 ],
  [ 0, 8, 1, 1, 1 ], [ 11, 0, 4, 1, 1 ], [ 2, 4, 4, 1, 1 ],
  [ 4, 0, 7, 1, 1 ], [ 10, 2, 0, 2, 1 ], [ 1, 6, 0, 2, 1 ],
  [ 3, 2, 3, 2, 1 ], [ 4, 0, 2, 3, 1 ], [ 11, 3, 0, 0, 2 ],
  [ 2, 7, 0, 0, 2 ], [ 4, 3, 3, 0, 2 ], [ 5, 1, 2, 1, 2 ],
  [ 7, 0, 1, 0, 3 ], [ 0, 0, 4, 0, 3 ] ]
gap> LengthsOfFactorizationsElementWRTNumericalSemigroup(415, S);
[ 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
  26, 27, 28, 32 ]

With the function MaximalDenumerant, we can compute the number of factorizations in \(\mathbf{Z}(415)\) with maximal length, that is, the maximal denumerant of \(415\) in \(S\).

gap> MaximalDenumerant(415,S);
1

References

Bryant, Hamblin, L. 2013. “The Maximal Denumerant of a Numerical Semigroup.” Semigroup Forum 86: 571–82. https://doi.org/10.1007/s00233-012-9448-5.

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.