Canonical ideal
Definition
Let \(S\) be a numerical semigroup with Frobenius number \(F(S)\) and \(E\) a relative ideal with Frobenius number of the ideal \(F(E)\). It is said that \(E\) is a canonical ideal if \(\tilde{E} = K(S)\), where \(\tilde{E} = E + F(S) - F(E)\) and \(K(S)\) denotes the standard canonical ideal of \(S\).
Examples
\(\circ\) Let \(S = \langle 4, 5, 7 \rangle = \{0, 4, 5, 7, \rightarrow \}\) and \(E = \{-15, -12, -11, -10, -8\} + S\). It is deduced that \(F(S) = 6\), \(F(E) = -9\) and \(K(S) = \{0, 3, 4, 5, 7, \rightarrow \}\). Then, \(\tilde{E} = E + F(S) - F(E) = E + 15\). It is concluded that \(\tilde{E} = \{0, 3, 4, 5, 7\} + S = K(S)\) and \(E\) is a 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 numerical semigroup \(S\), the function CanonicalIdeal and CanonicalIdealOfNumericalSemigroup return the standard canonical ideal of \(S\).
gap> K := CanonicalIdeal(S);
<Ideal of numerical semigroup>
gap> CanonicalIdeal(S) = CanonicalIdealOfNumericalSemigroup(S);
trueTo check if a relative ideal is a canonical ideal, we can use the functions IsCanonicalIdeal and IsCanonicalIdealOfNumericalSemigroup.
gap> IsCanonicalIdeal(IS);
false
gap> IsCanonicalIdeal(IS) = IsCanonicalIdealOfNumericalSemigroup(IS);
trueIn this case, \(S = K(S)\) (this happens if, and only if, \(S\) is symmetric).
References
https://gap-packages.github.io/ numericalsgps.