Standard canonical ideal
Definition
Let \(S\) be a numerical semigroup and \(F(S)\) the Frobenius number of \(S\). The standard canonical ideal of \(S\) is defined as
\[ K(S) = \{x \in \mathbb{Z} ~ | ~ F(S) - x \not \in S\}. \]
It can be proven that the standard canonical ideal is a relative ideal of \(S\) and \(S \subseteq K(S) \subseteq \mathbb{N}\).
Examples
\(\circ\) Let \(S = \langle 4, 9, 10 \rangle = \{0, 4, 8, 9, 10, 12, 13, 14, 16, \rightarrow \}\). The Frobenius number of \(S\) is \(F(S) = 15\), thus \(K(S) = \{x \in \mathbb{Z} ~ | ~ 15 - x \not \in S\}\). The set of gaps of \(S\) is \(G(S) = \{1, 2, 3, 5, 6, 7, 11, 15\}\), then if we compute \(15 -g\) for every \(g \in G(S)\), we have that \(K(S) = S \cup \{4, 8, 9, 10, 12, 13, 14\} = S\). It can be proven that \(S = K(S)\) if, and only if, \(S\) is symmetric.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 5, 7, 13, 16 \rangle\), in GAP:
gap> S := NumericalSemigroup(5, 7, 13, 16);
<Numerical semigroup with 4 generators>Given a numerical semigroup, the functions CanonicalIdeal and CanonicalIdealOfNumericalSemigroup return the standard canonical ideal of the numerical semigroup.
gap> K := CanonicalIdeal(S);
<Ideal of numerical semigroup>
gap> CanonicalIdeal(S) = CanonicalIdealOfNumericalSemigroup(S);
trueOn the other hand, given a relative ideal \(E\) of a numerical semigroup \(S\), the functions IsCanonicalIdeal and IsCanonicalIdealOfNumericalSemigroup return true or false depending on whether the relative ideal \(E\) is a translation of the canonical ideal, that is, if there exists \(x \in \mathbb{Z}\) such that \(E = x + K(S)\). In this case, it is said that \(E\) is a canonical ideal of \(S\).
gap> IsCanonicalIdeal(K);
true
gap> K2 := 3 + K;
<Ideal of numerical semigroup>
gap> IsCanonicalIdeal(K2);
true
gap> IsCanonicalIdeal(K2) = IsCanonicalIdealOfNumericalSemigroup(K2);
trueReferences
https://gap-packages.github.io/ numericalsgps.