M-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 \(S\). For \(x,y \in \mathbb{Z}\), it is defined the following partial order:
\[ x \preceq_S y \Longleftrightarrow x \le_S y ~~ \text{and} ~~ ord(y) = ord(x) + ord(y-x), \] where \(ord(\cdot)\) denotes the order of the respective element in \(S\). It is said that \(S\) is \(M-\)pure if the maximal elements of \(Ap(S,m)\) with respect to \(\preceq_S\) have all the same order.
It can be proven that if \(S\) is \(M-\)pure, then is pure.
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
\[ Ap(S,4) := \{s \in S ~ | ~ s - 4 \not \in S\} = \{0, 21, 10, 11\}. \]
The set of factorizations of each element in \(Ap(S,4) \setminus \{0\}\) is
\[ \mathbf{Z}(10) = \{(0,1,0)\}, ~~ \mathbf{Z}(11) = \{(0,0,1)\}, ~~ \mathbf{Z}(21) = \{(0,1,1)\}. \] Therefore, since \(10 \le_S 21\) and \(11 \le 21\) and \(ord(21) = ord(10) + ord(11)\), it is concluded that \(w = 21\) is the unique maximal element in \(Ap(S,4)\) with respect to \(\preceq_S\) and consequently \(S\) is \(M-\)pure.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 50, 99, 111, 119, 125, 139, 141, 146 \rangle\), in GAP:
gap> S := NumericalSemigroup(50, 99, 111, 119, 125, 139, 141, 146);
<Numerical semigroup with 8 generators>Given a numerical semigroup \(S\), the functions IsMpure and IsMpureNumericalSemigroup return true if \(S\) is \(M-\)pure and false otherwise.
gap> IsMpure(S);
true
gap> IsMpureNumericalSemigroup(S) = IsMpure(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.