Almost canonical ideal
Definition
Let \(S\) be a numerical semigroup with Frobenius number \(F(S)\) and genus \(g(S)\), and let \(E\) be a proper ideal with Frobenius number ideal \(F(E)\) and type ideal \(t(E)\). If \(\tilde{E} = E + F(S) - F(E)\), it is said that \(E\) is an almost canonical ideal if \(g(\tilde{E}) + m(\tilde{E}) = F(S) - g(S) + t(E)\), where \(g(\tilde{E})\) and \(m(\tilde{E})\) refers to the genus of an ideal and the multiplicity of an ideal, respectively.
If \(A - B = \{z \in \mathbb{Z} ~ | ~ z + B \subseteq A \}\) for \(A,B \subseteq \mathbb{N}\), it can be proven that \(E\) is an almost canonical ideal if, and only if, \(\tilde{E} - M = K(S) - M\), where \(K(S)\) is the standard-canonical ideal and \(M\) is the maximal ideal of \(S\).
Examples
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle = \{0, 5, 7, 9, 10, 12, 14, \rightarrow \}\) and \(E = \{10, 12, 14\} + S\). With simple calculations it is obtained that \(E \subseteq S\) (is a proper ideal), \(F(S) = 13, g(S) = 8\), \(F(E) = 18\) and \(t(E) = 3\). On the other hand, \(\tilde{E} = E + F(S) - F(E) = E - 5\), \(g(\tilde{E}) = 4\) and \(m(\tilde{E}) = 5\). Finally, \(F(S) - g(S) + t(E) = 13 - 8 + 3 = 8 \ne 9 = g(\tilde{E}) + m(\tilde{E})\). To sum up, \(E\) is not an almost canonical ideal.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 30, 43, 45, 50 \rangle\), \(I = \{30, 135, 140, 267\}\) and \(IS = I + S\), in GAP:
gap> S := NumericalSemigroup(30, 43, 45, 50);
<Numerical semigroup with 4 generators>
gap> I := [30, 135, 140, 267];
[ 30, 135, 140, 267 ]
gap> IS := I + S;
<Ideal of numerical semigroup>Given a proper ideal \(IS\), the function IsAlmostCanonical returns true or false depending on whether \(IS\) is an almost canonical ideal or not.
gap> IsAlmostCanonical(IS);
falseThe functions CanonicalIdeal and MaximalIdeal returns the standard canonical ideal and the maximal ideal of a given numerical semigroup, respectively.
gap> K := CanonicalIdeal(S);
<Ideal of numerical semigroup>
gap> M := MaximalIdeal(S);
<Ideal of numerical semigroup>
gap> IS2 := IS + FrobeniusNumber(S) - FrobeniusNumber(IS);
<Ideal of numerical semigroup>
gap> IS2 - M = K - M;
falseReferences
https://gap-packages.github.io/ numericalsgps.