Delta set of an element
Definition
Let \(S\) be a numerical semigroup, let \(s \in S\) and let \(\mathbf{L}(s)\) be the set of lengths of \(s\) in \(S\). Let us assume that \(\mathbf{L}(s) = \{l_1, l_2, \ldots, l_k\}\) with \(l_1 < l_2 < \cdots < l_k\). It is defined the Delta set of \(s\) in \(S\) as
\[ \Delta(s) = \{l_2 - l_1, \ldots, l_k - l_{k-1}\}, \]
and if \(k = 1\), \(\Delta(s) = \emptyset\). From the Delta set of an element it is defined the Delta set of a numerical semigroup.
Examples
\(\circ\) Let \(S = \langle 6, 13, 20 \rangle\), \(s = 72\) and \(\mathbf{Z}(s)\) the set of factorizations of \(s\). It is easy to deduce that \(\mathbf{Z}(72) = \{(12,0,0), (0,4,1), (1,2,2), (2,0,3)\}\). From the set of factorizations, we have \(\mathbf{L}(72) = \{5, 12\}\). Finally, \(\Delta(72) = \{7\}\).
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 25, 37, 38, 59 \rangle\), in GAP:
gap> S := NumericalSemigroup(25, 37, 38, 59);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\) and an element \(n \in S\), the functions DeltaSet and DeltaSetOfFactorizationsElementWRTNumericalSemigroup return the delta set of \(n\) in \(S\). If we only give a numerical semigroup \(S\) in DeltaSet, then it computes the Delta set of \(S\).
gap> DeltaSet(S, 612);
[ 1 ]
gap> DeltaSet(612, S);
[ 1 ]
gap> DeltaSetOfFactorizationsElementWRTNumericalSemigroup(612, S);
[ 1 ]
gap> DeltaSet(S);
[ 1, 2 ]The function Factorizations returns the factorizations of an element with respect to a numerical semigroup.
gap> Factorizations(S, 612);
[ [ 23, 1, 0, 0 ], [ 20, 2, 1, 0 ], [ 17, 3, 2, 0 ], [ 14, 4, 3, 0 ],
[ 11, 5, 4, 0 ], [ 8, 6, 5, 0 ], [ 5, 7, 6, 0 ], [ 2, 8, 7, 0 ],
[ 13, 0, 6, 1 ], [ 10, 1, 7, 1 ], [ 7, 2, 8, 1 ], [ 4, 3, 9, 1 ],
[ 1, 4, 10, 1 ], [ 2, 12, 0, 2 ], [ 0, 0, 13, 2 ], [ 10, 5, 0, 3 ],
[ 7, 6, 1, 3 ], [ 4, 7, 2, 3 ], [ 1, 8, 3, 3 ], [ 12, 0, 2, 4 ],
[ 9, 1, 3, 4 ], [ 6, 2, 4, 4 ], [ 3, 3, 5, 4 ], [ 0, 4, 6, 4 ],
[ 5, 2, 0, 7 ], [ 2, 3, 1, 7 ] ]On the other hand, the function LengthsOfFactorizationsElementWRTNumericalSemigroup returns the set of lengths of an element with respect to a numerical semigroup.
gap> I := LengthsOfFactorizationsElementWRTNumericalSemigroup(612, S);
[ 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 ]As we can see, \(l_i - l_{i-1} = 1\) for all \(i = 1, 2, \ldots, |\mathbf{L}(612)|\). Given only a list of integers \(I\), the function DeltaSet returns the set of differences of consecutive elements in the list.
gap> DeltaSet(I);
[ 1 ]\(\diamond\) Let \(S = \langle 7, 9, 13, 24 \rangle\), in GAP:
gap> S := NumericalSemigroup(7, 9, 13, 24);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\) and an integer \(n\), the function DeltaSetListUpToElementWRTNumericalSemigroup returns a list with the Delta sets of the integers up to (and including) \(n\). If an integer is not in \(S\), the corresponding Delta set is empty.
gap> DeltaSetListUpToElementWRTNumericalSemigroup(100, S);
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[ 1, 2 ], [ 1, 2 ] ]On the other hand, given a numerical semigroup \(S\) and a non-negative integer \(n\), the function DeltaSetUnionUpToElementWRTNumericalSemigroup returns the union of the Delta sets of the elements of \(S\) up to (and including) \(n\).
gap> DeltaSetUnionUpToElementWRTNumericalSemigroup(1000, S);
[ 1 .. 3 ]References
https://gap-packages.github.io/ numericalsgps.