Star closure
Definition
Let \(S\) be a numerical semigroup, let \(E\) be a relative ideal of \(S\) and \(Is = \{E_1, \ldots, E_n\}\) a set of relative ideals of \(S\). It is defined the star closure of \(E\) with respect to \(Is\), denoted by \(St(E)\), as
\[ st(E) = (S - (S - E)) \bigcap_{k \in \{1, \ldots, n\}} (E_k - (E_k - E)), \]
where for any \(A,B \subseteq \mathbb{Z}\),
\[ A - B = \{z \in \mathbb{Z} ~ | ~ z + B \subseteq A\}. \]
Examples
\(\circ\) Let \(S = \langle 4, 6, 11 \rangle = \{ 0, 4, 6, 8, 10, 11, 12, 14, \rightarrow \}\), \(I = \{-2, 3\} + S\) and \(Is = \{J,K\}\), where \(J = \{-10, -7\} + S\) and \(K = \{1, 2\} + S\). With few computations,
\[ I = \{-2, 2, 3, 4, 6, \rightarrow\}, ~~ J = \{ -10, -7, -6, -4, \rightarrow\}, ~~ K = \{1, 2, 5, \rightarrow\}. \]
On the other hand, taking into account that \(\{16, \rightarrow \} \subseteq S - I \subseteq \{2, \rightarrow\}\), \(\{-2,\rightarrow\} \subseteq J - I \subseteq \{-8, \rightarrow\}\) and \(\{7, \rightarrow\} \subseteq K - I \subseteq \{3, \rightarrow \}\), it is obtained that
\[ S - I = \{8, 12, 13, 14, 16, \rightarrow\}, ~~ J - I = \{-5, -4, -2, \rightarrow\}, ~~ K - I = \{3, 4, 7, \rightarrow\}. \]
Finally, with the same argument as above,
\[ S - (S - I) = \{-2, 2, 3, 4, 6, \rightarrow\}, ~~ J - (J - I) = \{-2, 1, \rightarrow\}, ~~ K - (K - I) = \{-2, 2, \rightarrow\}, \]
and
\[ St(S) = \{-2, 2, 3, 4, 6, \rightarrow\} \cap \{-2, 1, \rightarrow\} \cap \{-2, 2, \rightarrow\} = \{-2, 2, 3, 4, 6, \rightarrow\}. \]
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 45, 53, 58, 86, 89, 109, 117 \rangle\), \(I = \{19, 23, 58, 59, 61, 85 \} + S\) and \(Is = \{J, K, L\}\) where \(J = \{ 2, 22, 26, 46, 52, 98\} + S, K = \{-9, -5, 17, 35, 83 \} + S, L = \{-6, 0, 41, 49, 93 \} + S\) , in GAP:
gap> S := NumericalSemigroup(45, 53, 58, 86, 89, 109, 117);
<Numerical semigroup with 7 generators>
gap> I := [19, 23, 58, 59, 61, 85] + S;
<Ideal of numerical semigroup>
gap> J := [2, 22, 26, 46, 52, 98] + S;
<Ideal of numerical semigroup>
gap> K := [-9, -5, 17, 35, 83] + S;
<Ideal of numerical semigroup>
gap> L := [-6, 0, 41, 49, 93] + S;
<Ideal of numerical semigroup>
gap> Is := [J,K,L];
[ <Ideal of numerical semigroup>, <Ideal of numerical semigroup>,
<Ideal of numerical semigroup> ]Given a numerical semigroup \(S\), a relative ideal \(I\) and a set of relative ideals \(Is\), the function StarClosureOfIdealOfNumericalSemigroup returns the star closure of \(I\) with respect to \(Is\).
gap> St := StarClosureOfIdealOfNumericalSemigroup(I, Is);
<Ideal of numerical semigroup>
gap> SmallElements(St);
[ 19, 23, 58, 59, 61, 64, 68, 72, 76, 77, 81, 85, 92, 98, 103, 104, 105, 106,
108, 109, 111, 112, 113, 114, 116, 117, 118, 119, 121, 122, 124, 125, 126,
128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157,
158, 159, 161 ]
gap> M := MinimalGenerators(St);
[ 19, 23, 58, 59, 61, 85, 92, 98, 118, 124, 131, 133, 141, 142, 146, 152, 155 ]The functions SmallElements and MinimalGenerators returns the set of small elements and thee set of minimal generators, respectively. Then, the star closure is \(St(I) = M + S\).
References
https://gap-packages.github.io/ numericalsgps.