The complexity of a numerical semigroup

TL;DR

This paper introduces algorithms to construct all ideal extensions of numerical semigroups, defines a complexity measure based on recursive ideal extensions, and computes semigroups with fixed multiplicity and complexity.

Contribution

It presents a novel recursive algorithm for building ideal extensions and a method to determine the complexity of numerical semigroups.

Findings

Algorithm to generate all ideal extensions of a given numerical semigroup.

Definition of the complexity of a numerical semigroup based on recursive ideal extensions.

Method to compute all semigroups with fixed multiplicity and complexity.

Abstract

Let $S$ and $\Delta$ be numerical semigroups. A numerical semigroup $S$ is an $\mathbf{I}(\Delta)$-{\it semigroup} if $S\backslash \{0\}$ is an ideal of $\Delta$. We will denote by $\mathcal{J}(\Delta)=\{S \mid S \text{ is an $\mathbf{I}(\Delta)$-semigroup} \}.$ We will say that $\Delta$ is {\it an ideal extension of } $S$ if $S\in \mathcal{J}(\Delta).$ In this work, we present an algorithm that allows to build all the ideal extensions of a numerical semigroup. We can recursively denote by $\mathcal{J}^0(\mathbb{N})=\mathbb{N},$ $\mathcal{J}^1(\mathbb{N})=\mathcal{J}(\mathbb{N})$ and $\mathcal{J}^{k+1}(\mathbb{N})=\mathcal{J}(\mathcal{J}^{k}(\mathbb{N}))$ for all $k\in \mathbb{N}.$ The complexity of a numerical semigroup $S$ is the minimun of the set $\{k\in \mathbb{N}\mid S \in \mathcal{J}^k(\mathbb{N})\}.$ In addition, we will give an algorithm that allows us to compute all the numerical semigroups with fixed multiplicity and complexity.