Irreducible numerical semigroups with multiplicity three and four

TL;DR

This paper studies the irreducibility of numerical semigroups with multiplicities three and four using Kunz-coordinates, providing a complete description of their minimal decompositions and applying the methodology to well-known families.

Contribution

It introduces a complete characterization of irreducible numerical semigroups with multiplicities three and four using Kunz-coordinates, including their minimal decompositions.

Findings

Complete description of irreducible semigroups with multiplicity 3 and 4.

Methodology applicable to families generated by generalized arithmetic progressions.

Conditions for irreducibility of specific well-known semigroup families.

Abstract

In this paper we analyze the irreducibility of numerical semigroups with multiplicity up to four. Our approach uses the notion of Kunz-coordinates vector of a numerical semigroup recently introduced in (Blanco-Puerto, 2011). With this tool we also completely describe the whole family of minimal decompositions into irreducible numerical semigroups with the same multiplicity for this set of numerical semigroups. We give detailed examples to show the applicability of the methodology and conditions for the irreducibility of well-known families of numerical semigroups as those that are generated by a generalized arithmetic progression.