Homogeneous numerical semigroup
Definition
Let \(S\) be a numerical semigroup with multiplicity \(m\) and let \(Ap(S,m)\) be the Apéry set of \(m\) in \(S\). It is said that \(S\) is homogeneous if for every element in \(Ap(S, m)\), its set of lengths has one element, that is, all its factorizations has the same length.
From the definition, if \(S\) has Apéry set of unique expression, then \(S\) is homogeneous.
Examples
\(\circ\) Let \(S = \langle 7, 12, 15 \rangle\). The multiplicity of \(S\) is \(m = 7\) and the Apéry set of \(m\) in \(S\) is
\[ Ap(S, 7) = \{0, 15, 30, 24, 39, 12, 27\}. \] The set of factorizations of each element in \(Ap(S,7) \setminus \{0\}\) is
\[ \mathbf{Z}(15) = \{(0,0,1)\}, ~~ \mathbf{Z}(30) = \{(0,0, 2)\}, ~~ \mathbf{Z}(24) = \{(0,2,0)\}, \]
\[ \mathbf{Z}(39) = \{(0,2,1)\}, ~~ \mathbf{Z}(10) = \{(0,1,0)\}, ~~ \mathbf{Z}(12) = \{(0,1,0)\}, ~~ \mathbf{Z}(27) = \{(0,1,1)\} \]
Since each element has a unique factorization, it is concluded that \(S\) is homogeneous.
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 10, 84, 89, 96, 112, 113, 131 \rangle\), in GAP:
gap> S := NumericalSemigroup(10, 84, 89, 96, 112, 113, 131);
<Numerical semigroup with 7 generators>Given a numerical semigroup \(S\), the function IsHomogeneousNumericalSemigroup returns true if \(S\) is homogeneous and false otherwise.
gap> IsHomogeneousNumericalSemigroup(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.