Eliahou number
Definition
Let \(S\) be a numerical semigroup minimally generated by \(P(S)\), let \(N(S)\) be the set of left elements with \(n(S) = |N(S)|\), depth and remainder \(q(S)\) and \(\rho(S)\) respectively, and threshold interval \(D\). If \(p_q\) denotes the minimal generators in \(D\) and \(d_q = |D| - p_q\), it is defined the Eliahou number of \(S\), denoted by \(E(S)\) or \(W_0(S)\), as the value
\[ E(S) = |P(S) \cap N(S)|n(S) - q(S)d_q + \rho(S). \]
If \(W(S)\) is the Wilf number of \(S\), it can be proven that \(W(S) \ge E(S)\). In particular, if \(E(S) \ge 0\), then \(S\) is a Wilf semigroup.
Examples
\(\circ\) Let \(S = \langle 14, 22, 23, 57, 61, 62, 63 \rangle\). The set of left elements of \(S\) is
\[ N(S) = \{0, 14, 22, 23, 28, 36, 37, 42, 44, 45, 46, 50, 51\}, \]
and the threshold is \(I = \{56, 57, \ldots, 69\}\). Therefore,
\[ |P(S) \cap N(S)| = |\{14, 22, 23\}| = 3, ~~ n(S) = |N(S)| = 13,~~ q(S) = \left \lceil \frac{56}{14} \right \rceil = 4, \]
\[ \rho(S) = 0, ~~ d_q = |D| - p_q = 14 - p_q = 10, \]
and the Eliahou number of \(S\) is
\[ E(S) = |P(S) \cap N(S)|n(S) - q(S)d_q + \rho(S) = 3 \cdot 13 - 4 \cdot 10 + 0 = -1 < 0. \]
On the other hand, \(W(S) = e(S)n(S) - C(S) = 35\). This example shows that there exists numerical semigroups such that \(E(S) < 0\) but \(W(S) > 0\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 16, 17, 41, 70 \rangle\), in GAP:
gap> S := NumericalSemigroup(16, 17, 41, 70);
<Numerical semigroup with 4 generators>Given a numerical semigroup \(S\), the functions EliahouNumber and TruncatedWilfNumberOfNumericalSemigroup compute the Eliahou number of \(S\).
gap> EliahouNumber(S);
81
gap> TruncatedWilfNumberOfNumericalSemigroup(S);
81The function WilfNumber returns the Wilf number given a numerical semigroup. In this case,
gap> WilfNumber(S);
81concluding that there exist numerical semigroups such that \(W(S) = E(S)\).
References
https://gap-packages.github.io/ numericalsgps.