Delta set of a numerical semigroup
Definition
Let \(S\) be a numerical semigroup minimally generated by \(P = \{n_1, \ldots, n_p\}\). It is defined the Delta set of \(S\), denoted by \(\Delta(S)\), as
\[ \Delta(S) = \bigcup_{s \in S} \Delta(s), \]
where \(\Delta(s)\) denotes the delta set of s in \(S\). If \(Betti(S)\) is the set of Betti elements of \(S\), it can be proven that
\[ \max \Delta(S) = \max \{\max \Delta(b) ~ | ~ b \in Betti(S)\}. \]
On the other hand, there exists \(N_S \in \mathbb{N}\) such that \(\Delta(s)\) is periodic from \(N_S\) and its period is equal to \(lcm(n_1, n_p)\).
Given a numerical semigroup \(S\), there is a relation between its Delta set \(\Delta(S)\), its catenary degree \(\mathbf{C}(S)\), its w-primality \(\omega(S)\) and its tame degree \(\mathbf{t}(S)\), which is
\[ \max \Delta(S) + 2 \le \mathbf{C}(S) \le \omega(S) \le \mathbf{t}(S). \]
Examples
\(\circ\) Let \(S = \langle 2, 3 \rangle\). If \(\nabla_s\) is not connected, where \(\nabla_s\) denotes the graph associated to s, then \(s = n + w\) with \(n \in \{2,3\}\) and \(w \in Ap(S,2)\). From the above argument, it is deduced that \(Betti(S) = \{6\}\), and its set of factorizations is \(\mathbf{Z}(6) = \{(3,0),(0,2)\}\). Thus, \(\Delta(6) = \{1\}\) and \(\max \Delta(S) = \{1\}\). Finally, \(\Delta(S) = \{1\}\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 13, 45, 60, 89 \rangle\), in GAP:
gap> S := NumericalSemigroup(13, 45, 60, 89);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\), the functions DeltaSet and DeltaSetOfNumericalSemigroup return the Delta set of \(S\). If we also give an element \(n \in S\) in DeltaSet, the function returns the Delta set of \(n\) in \(S\).
gap> DeltaSet(S);
[ 1, 2, 3, 4, 6, 7, 11 ]
gap> DeltaSet(S, 314);
[ 1, 2, 4 ]Given a numerical semigroup \(S\), the function DeltaSetPeriodicityBoundForNumericalSemigroup returns the bound where the periodicity starts for Delta sets of the elements in \(S\), \(N_S\). On the other hand, the function DeltaSetPeriodicityStartForNumericalSemigroup computes the actual element were the periodicity stars for Delta sets of the elements in \(S\).
gap> DeltaSetPeriodicityBoundForNumericalSemigroup(S);
10065
gap> DeltaSetPeriodicityStartForNumericalSemigroup(S);
496