Frobenius number of an ideal
Definition
Let \(S\) be a numerical semigroup and \(E\) a relative ideal of \(S\). The Frobenius number of \(E\) is defined as the maximum of the set \(\mathbb{Z} \setminus E\), and it is denoted as \(F(E)\).
It can be proven from the definition of relative ideal that the set \(\mathbb{Z} \setminus E\) always has a maximum. Unlike the Frobenius number of a numerical semigroup (other than \(\mathbb{N}\)), \(F(E)\) can be negative.
Examples
\(\circ\) Let \(S\) a numerical semigroup, \(F(S)\) its Frobenius number and \(E\) a relative ideal of \(S\). Let
\[ \tilde{E} = E + (F(S) - F(E)) = \{e + F(S) - F(E) ~ | ~ e \in E \}. \]
The maximum of \(\mathbb{Z} \setminus \tilde{E}\) is the maximum of \(\mathbb{Z} \setminus E\) adding \(F(S) - F(E)\), that means, \(F(\tilde{E}) = F(E) + F(S) - F(E) = F(S)\). To sum up, every relative ideal \(E\) of \(S\) is equivalent to a relative ideal with same Frobenius number of \(S\).
\(\circ\) Let \(S\) a numerical semigroup, \(F(S)\) its Frobenius number and \(K(S)\) its standard canonical ideal. By definition, \(x \in K(S)\) if, and only if, \(F(S) - x \not \in S\). Thus, \(F(S) \not \in K(S)\). Furthermore, if \(x > F(S)\), \(F(S) - x < 0\) and \(x \in K(S)\). In conclusion, \(F(K(S)) = F(S)\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 15, 17, 23 \rangle\), \(I = \{-12, -5, 4, 10\}\) and \(IS = I + S\), in GAP:
gap> S := NumericalSemigroup(15, 17, 23);
<Numerical semigroup with 3 generators>
gap> I := [-12, -5, 4, 10];
[ -12, -5, 4, 10 ]
gap> IS := I + S;
<Ideal of numerical semigroup>The functions FrobeniusNumber and FrobeniusNumberOfIdealOfNumericalSemigroup return the Frobenius number of a given relative ideal. The function FrobeniusNumber also returns the Frobenius number of a given numerical semigroup.
gap> FrobeniusNumber(IS);
47
gap> FrobeniusNumber(IS) = FrobeniusNumberOfIdealOfNumericalSemigroup(IS);
true
gap> FrobeniusNumber(S);
88References
https://gap-packages.github.io/ numericalsgps.