Irreducibility of Arf Numerical Semigroups

TL;DR

This paper explores the properties of Arf numerical semigroups, introducing concepts like Arf special gaps and Arf-irreducibility, along with algorithms and generators for their systematic analysis.

Contribution

It introduces the notion of Arf special gaps and Arf-irreducibility, providing algorithms and generator systems for these semigroups.

Findings

Defined Arf special gaps and Arf-irreducibility.

Developed an algorithm to compute all Arf special gaps.

Established systems of generators for Arf numerical semigroups.

Abstract

In this paper, we introduce the concept of Arf special gaps of an Arf numerical semigroup, and an algorithm for computing all Arf special gaps of a given Arf numerical semigroup. We introduce the concept of Arf-irreducible numerical semigroups, and draw conclusions about all these concepts. We give a system of generators for the Frobenius variety and variety of families of Arf numerical semigroups. Moreover, we obtain the minimal Arf system of generators of a given Arf numerical semigroup.