Ratliff-Rush closure of a proper ideal
Definition
Let \(S\) be a numerical semigroup and \(E\) a proper ideal of \(S\). It is defined the Ratliff-Rush closure of \(E\), denoted by \(RR(E)\) as
\[ RR(E) = S \cap \bigcup_{n \in \mathbb{N}} (n+1)I - nI, \]
where for any \(A,B \subseteq \mathbb{Z}\),
\[ A - B = \{z \in \mathbb{Z} ~ | ~ z + B \subseteq A\}, ~~~~ nA = \{a_1 + \cdots + a_n ~ | ~ a_1, \ldots, a_n \in A\}. \]
It can be proven that \(nI - (n-1)I \subseteq (n+1)I - nI\) for all \(n \in \mathbb{N} \setminus \{0\}\) and that the chain
\[ I \subseteq 2I - I \subseteq 3I - 2I \subseteq \cdots \subseteq (n+1)I - nI \subseteq \cdots \subseteq \mathbb{Z}, \]
stabilizes. In general, \(nI - (n-1)I = (n+1)I - nI\) does not imply that \((n+1)I - nI = (n+2)I - (n+1)I\).
Examples
\(\circ\) Let \(I\) a proper ideal of a certain numerical semigroup and let \(r(I)\) be the reduction number of \(I\), that is, the minimum \(h \ge 1\) such that \((h + 1)I = m(I) + hI\), where \(m(I)\) is the multiplicity of \(I\). Let \(h \ge r(I)\) and let us prove that \((h+1)I - hI = (h+2)I - (h+1)I\). Since the inclusion \(\subseteq\) always holds, let us prove the another inclusion. Given \(z \in (h+2)I - (h+1)I\), by definition \(z + (h+1)I \subseteq (h+2)I\), then
\[ z + (h+1)I \subseteq z + (h+2)I \Longleftrightarrow z + (r(I) + 1)I + (h - r(I))I \subseteq (r(I) + 1)I + (h - r(I) + 1)I \]
\[ \Longrightarrow z + m(I) + r(I)I + (h - r(I))I \subseteq m(I) + r(I)I + (h - r(I) + 1)I \Longrightarrow z + hI \subseteq (h+1)I, \]
concluding that \(z \in (h+1)I - hI\). From this demonstration, we have that the Ratliff-closure of any relative ideal is
\[ RR(I) = S \cap ( (r(I) + 1)I - r(I)I ). \]
\(\circ\) Let \(S = \langle 3, 5, 7 \rangle\) and \(I = \{2, 4\} + S = \{2, 4, 5, 7, \rightarrow \}\). Clearly, \(m(I) = 2\), then we are looking for the smaller \(h \in \mathbb{N} \setminus \{0\}\) such that \((h+1)I = 2 + hI\). If we take \(h = 1\),
\[ 2I = I + I = \{4, 6, \rightarrow\} \ne \{4, 6, 7, 9, \rightarrow \} = 2 + I. \]
For \(h = 2\),
\[ 3I = (I + I) + I = \{6, 8, \rightarrow\} = 2 + 2I \]
Then, the reduction number of \(I\) is \(r(I) = 2\) and its Ratliff-Rush closure is
\[ RR(I) = S \cap (3I - 2I) = S \cap \{2, 4, \rightarrow\} = \{5, \rightarrow \}, \] but on the other hand,
\[ S \cap (2I - I) = \{5, \rightarrow \}. \]
The least integer \(n\) such that \(S \cap ((n+1)I - nI)\) is the Ratliff-Rush closure is called the Ratliff-Rush number of \(I\) and it is denoted by \(R(I)\). From what we have seen above, \(R(I) \le r(I)\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 50, 54, 78, 115, 116, 119 \rangle\), in GAP:
gap> S := NumericalSemigroup(50, 54, 78, 115, 116, 119);
<Numerical semigroup with 6 generators>
gap> I := [50, 78, 119] + S;
<Ideal of numerical semigroup>Given a relative ideal \(I\) of a numerical semigroup, the functions RatliffRushClosure and RatliffRushClosureOfIdealOfNumericalSemigroup return the Ratliff-Rush closure of \(I\).
gap> RR := RatliffRushClosure(I);
<Ideal of numerical semigroup>
gap> L := MinimalGenerators(RR);
[ 50, 78, 119, 162, 231, 345 ]Given a relative ideal, the function MinimalGenerators returns its minimal set of generators. Then, the Ratliff-Rush closure of \(I\) is \(RR(I) = L + S\).
References
https://gap-packages.github.io/ numericalsgps.