Relative ideal
Definition
Let \(S\) be a numerical semigroup. A subset \(E\) of \(\mathbb{Z}\) is a relative ideal of \(S\) if \(S + E = \{s + e ~ | ~ s \in S, e \in E \} \subseteq E\) and there exists \(s \in S\) such that \(s + E = \{s + e ~ | ~ e \in E \}\subseteq S\).
Two relative ideals \(E\) and \(H\) are said to be equivalent if there exists \(x \in \mathbb{Z}\) such that \(E = x + H = \{x + h ~ | ~ h \in H \}\). As a consequence, every relative ideal is equivalent to some proper ideal.
It is said that a subset \(I = \{e_1, \ldots, e_h\} \subseteq E\) is a system of generators of \(E\) if \(E = I + S\). It can be proven that for every relative ideal \(E\) of \(S\) there exists a unique minimal set of generators, which is \(A = E \setminus (M + E)\), where \(M = S \setminus \{0\}\) is the maximal ideal of \(S\).
Examples
\(\circ\) Let \(S = \langle 3, 8, 10 \rangle = \{0, 3, 6, 8, \rightarrow \}\) and \(E = \langle 3 \rangle\) the monoid generated by \(A = \{3\}\) in \(\mathbb{N}\). \(E\) is not a relative ideal since \(S + E \subsetneq E\), for example \(3 + 8 = 11 \not \in E\) but \(3 \in E\) and \(8 \in S\).
\(\circ\) Let \(S\) a numerical semigroup, \(k \in \mathbb{N} \setminus \{0\}\) and \(I = \{i_1, i_2, \ldots, i_k\}\) a subset of \(\mathbb{Z}\) arbitrary but fixed. Then, the set \(E = I + S\) is a relative ideal of \(S\). The first condition is satisfied because \(S\) is a numerical semigroup. moreover, if we take \(w = \min \{0, i_1, i_2, \ldots, i_k\}\), defining \(s = C(S) - w\), where \(C(S)\) denotes the conductor of \(S\), we have that \(s \in S\) and \(s + E \subseteq S\). In conclusion, \(E\) is a relative ideal.
Examples with GAP
The following examples are made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S\) a numerical semigroup and \(I\) a list of integers. As we know, the subset \(E = I + S\) is a relative ideal of \(S\), the function IdealOfNumericalSemigroup returns the ideal generated by \(I\) and \(S\).
gap> S := NumericalSemigroup(12,17,20);
<Numerical semigroup with 3 generators>
gap> I := [-5, -4, 13];
[ -5, -4, 13 ]
gap> IS := IdealOfNumericalSemigroup(I,S);
<Ideal of numerical semigroup>It is also possible to generate it with the + operator.
gap> J := I + S;
<Ideal of numerical semigroup>
gap> J = IS;
trueIn the case where we have an ideal \(I\), the function IsIdealOfNumericalSemigroup checks whether is a relative ideal of a numerical semigroup or not.
gap> T := [1..5] + NumericalSemigroup(3,8);
<Ideal of numerical semigroup>
gap> IsIdealOfNumericalSemigroup(T);
trueOn the other hand, if \(I\) is a relative ideal for some numerical semigroup \(S\), the function AmbientNumericalSemigroupOfIdeal returns the numerical semigroup.
gap> T := AmbientNumericalSemigroupOfIdeal(IS);
<Numerical semigroup with 3 generators>
gap> S = T;
true\(\diamond\) Given an ideal \(IS\), the functions MinimalGenerators, MinimalGeneratingSystem and MinimalGeneratingSystemOfIdealOfNumericalSemigroup return the minimal system of generators of \(IS\).
gap> S := NumericalSemigroup(6, 14, 22, 37, 53);
<Numerical semigroup with 5 generators>
gap> IS := [-7, -3, -1, 4, 12, 22]+S;
<Ideal of numerical semigroup>
gap> MinimalGenerators(IS);
[ -7, -3, 4, 12 ]
gap> MinimalGenerators(IS) = MinimalGeneratingSystem(IS);
true
gap> MinimalGenerators(IS) = MinimalGeneratingSystemOfNumericalSemigroup(IS);
trueOn the other hand, the functions Generators and GeneratorsOfIdealOfNumericalSemigroup return a system of generators of a relative ideal. It may not be minimal.
gap> Generators(IS);
[ -7, -3, -1, 4, 12, 22 ]
gap> Generators(IS) = GeneratorsOfIdealOfNumericalSemigroup(IS);
true\(\diamond\) Given a relative ideal \(IS\) and an integer \(n\), the function BelongsToIdealOfNumericalSemigroup returns true or false depending on whether \(n \in IS\) or not. The expression \(n ~~ in ~~ IS\) can also be used.
gap> S := NumericalSemigroup(7, 41, 79, 113, 150);
<Numerical semigroup with 5 generators>
gap> IS := [-40, -21, -10, 2, 3] + S;
<Ideal of numerical semigroup>
gap> BelongsToIdealOfNumericalSemigroup(114, IS);
true
gap> 114 in IS;
trueGiven a relative ideal \(IS\) and a positive integer \(r\), the function ElementNumber_IdealOfNumericalSemigroup returns the \(r-\)th element in \(IS\). The expression \(IS[r]\) can also be used.
gap> ElementNumber_IdealOfNumericalSemigroup(IS, 35);
29
gap> IS[35];
29If we have a list of positive integers \(L\), the expression \(IS\{L\}\) returns the list \([IS[k] ~ | ~ k \in L]\).
gap> IS{[2, 5, 41, 152]};
[ -33, -19, 36, 150 ]Conversely, if \(IS\) is a relative ideal and \(r\) an integer, the function NumberElement_IdealOfNumericalSemigroup returns the position of \(r\) in \(IS\). Returns fail if \(r \not \in IS\).
gap> NumberElement_IdealOfNumericalSemigroup(IS, 114);
116
gap> NumberElement_IdealOfNumericalSemigroup(IS, -4);
failReferences
https://gap-packages.github.io/ numericalsgps.