Principal ideal
Definition
Let \(S\) be a numerical semigroup and \(E \subseteq \mathbb{Z}\). It is said that \(E\) is a principal ideal of \(S\) if it is a proper ideal of the form \(E = s + S\) for some \(s \in S \setminus \{0\}\).
It can be proven that \(E\) is a principal ideal of \(S\) if, and only if, \(m(E) \in S\) and \(E = m(E) + S\), where \(m(E)\) denotes the multiplicity ideal of \(E\).
Examples
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle = \{0, 5, 7, 9, 10, 12, 14, \rightarrow\}\) and \(E = \{12, 17, 19, 21, 22, 24, 26, \rightarrow\}\), let us prove that \(E = 12 + S\).
\(\subseteq\) Let \(x \in E\). If \(x \ge 26\), then \(x - 12 \ge 14\) and \(x-12 \in S\), which means that \(x \in (12 + S)\). If \(x < 26\), then \(x \in \{12, 17, 19, 21, 22, 24\} = 12 + \{0, 5, 7, 9, 10, 12\} \subseteq (12 + S)\).
\(\supseteq\) Let \(x = 12 + s\) for some \(s \in S\). If \(x > 26\), clearly \(x \in E\). If \(x < 26\), then \(s < 14\) and \(s \in \{0, 5, 7, 9, 10, 12\}\), where in each case it holds \(x \in E\).
To sum up, \(E\) is a principal ideal of \(S\).
Examples with GAP
Nowadays, there are no functions in package NumericalSgps related to principal ideal. However, given a numerical semigroup \(S\) and a relative ideal \(I\), the following function returns true if \(I\) is a principal ideal of \(S\).
gap> IsPrincipalIdeal := function(S,I)
> local m;
> if not IsNumericalSemigroup(S) then
> Error("First argument must be a Numerical Semigroup");
> fi;
> if not IsIdealOfNumericalSemigroup(I) then
> Error("Second argument must be a relative ideal of the first argument");
> fi;
> if not AmbientNumericalSemigroupOfIdeal(I) = S then
> Error("Second argument must be a relative ideal of the first argument");
> fi;
> m := Minimum(I);
> if m in S and Length(MinimalGenerators(I)) = 1 then
> return true;
> fi;
> return false;
>
> end;
function( S, I ) ... end\(\diamond\) Let \(S = \langle 33, 28, 51, 64, 90 \rangle\) and \(I = \{151,184, 207, 217, 273, 291\} + S\), in GAP:
gap> S := NumericalSemigroup(33, 28, 51, 64, 90);
<Numerical semigroup with 5 generators>
gap> I := [151, 184, 207, 217, 273, 291] + S;
<Ideal of numerical semigroup>Applying the function defined above,
gap> IsPrincipalIdeal(S,I);
trueThen, \(I = 151 + S\) with \(s = 151 \in S\).
References
https://gap-packages.github.io/ numericalsgps.