Maximum sparse ideal
Definition
Let \(S\) be a numerical semigroup with genus \(g(S)\) and Frobenius number \(F(S)\), and let \(E\) be a proper ideal of \(S\) with Frobenius number ideal \(F(E)\). It is said that \(E\) is a maximum sparse ideal if it satisfies the equality \(F(E) = 2g(S) - 1 + |S \setminus E|\).
If the multiplicity of a relative ideal is denoted by \(m(E)\) and the genus of a relative ideal by \(g(E)\), it can be proven that the equality is equivalent to \(F(S) + 1 - g(S) = g(\tilde{E}) + m(\tilde{E})\), where \(\tilde{E} = E + F(S) - F(E)\).
Examples
\(\circ\) Let \(S = \langle 4, 9 \rangle\) and the proper ideal \(E = \{F(S) + 1, \rightarrow \}\) (in fact, is the conductor ideal of \(S\)). In this case, \(E = \tilde{E}\), and it holds \(F(S) = 11, ~ g(S) = 6, ~ g(E) = 0\) and \(m(E) = F(S) + 1\). Then, \(F(S) + 1 - g(S) = 6 < 12 = m(\tilde{E}) + g(\tilde{E})\). Hence, \(E\) is not a maximum sparse ideal.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 13, 32, 79, 81, 101 \rangle\) and \(IS = \{26, 45, 58, 94, 111\} + S\), in GAP:
gap> S := NumericalSemigroup(13, 32, 79, 81, 101);
<Numerical semigroup with 5 generators>
gap> IS := [26, 45, 58, 94, 111] + S;
<Ideal of numerical semigroup>It can be proven that a proper ideal \(E\) is a maximum sparse ideal if, and only if, \(\tilde{E} = K(S)\), where \(K(S)\) is the standard canonical ideal of \(S\). Therefore, if \(E\) is a maximum sparse ideal, then is a canonical ideal of \(S\) such that \(E \subseteq S\). Given a relative ideal \(IS\), the function IsCanonicalIdeal returns true or false depending on whether \(IS\) is a canonical ideal or not.
gap> IsSubset(0+S, IS);
true
gap> IsCanonicalIdeal(IS);
falseTo sum up, \(IS\) is not a maximum sparse ideal.
References
https://gap-packages.github.io/ numericalsgps.