The lattice of ideals of a numerical semigroup and its Frobenius restricted variety associated

Abstract

Let $\Delta$ be a numerical semigroup. In this work we show that $\mathcal{J}(\Delta) =\{I\cup\nobreak\{0\} I is an ideal of \Delta\}$ is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set $\mathcal{J}_a(\Delta)=\{S\in\mathcal{J}(\Delta) \max(\Delta\backslash S)=a\}$ for a given $a\in\Delta.$ As a consequence, we obtain another algorithm that computes all the elements of $\mathcal{J}(\Delta)$ with a fixed genus.

Keywords

• numerical semigroup
• ideal
• frobenius restricted variety
• embedding dimension
• frobenius number
• restricted frobenius number
• genus
• multiplicity
• arf numerical semigroup
• saturated semigroup