Pure numerical semigroup
Definition
Let \(S\) be a numerical semigroup with multiplicity \(m\), let \(Ap(S,m)\) be the Apéry set of \(m\) in \(S\) and let \(\le_S\) be the relation order of \(S\). It is said that \(S\) is pure if the maximal elements of \(Ap(S,m)\) with respect to \(\le_S\) have all the same order.
Examples
\(\circ\) Let \(S = \langle 4, 10, 11 \rangle = \{0, 4, 8, 10, 11, 12, 14, 15, 16, 18, \rightarrow\}\). The multiplicity of \(S\) is \(m = 4\) and the Apéry set of \(m\) in \(S\) is
\[ Ap(S,4) := \{s \in S ~ | ~ s - 4 \not \in S\} = \{0, 21, 10, 11\}. \]
Since \(10 \le_S 21\) and \(11 \le 21\), it is concluded that \(w = 21\) is the unique maximal element in \(Ap(S,4)\) with respect to \(\le_S\) and consequently \(S\) is pure.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 17, 30, 35, 50, 72, 88 \rangle\), in GAP:
gap> S := NumericalSemigroup(17, 30, 35, 50, 72, 88);
<Numerical semigroup with 6 generators>Given a numerical semigroup \(S\), the functions IsPure and IsPureNumericalSemigroup return true if \(S\) is pure and false otherwise.
gap> IsPure(S);
true
gap> IsPureNumericalSemigroup(S) = IsPure(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.