Abstract

Let S=⟨a1,…,ap⟩ be a numerical semigroup, let s∈S and let Z(s) be its set of factorizations. The set of lengths is denoted by L(s)={L(x1,⋯,xp)∣(x1,⋯,xp)∈Z(s)}, where L(x1,⋯,xp)=x1+⋯+xp. The following sets can then be defined: W(n)={s∈S∣∃x∈Z(s)suchthatL(x)=n}, ν(n)=⋃s∈W(n)L(s)={l1<l2<⋯<lr} and Δν(n)={l2−l1,…,lr−lr−1}. In this paper, we prove that the function Δν:N→P(N) is almost periodic with period lcm(a1,ap).

Highlights

  • A numerical semigroup is a finitely generated subsemigroup of the set of nonnegative integers N, such that the group generated by it is the set of all integers Z

  • Every numerical semigroup is finitely generated and their elements might be expressed in different ways as a linear combination with non-negative integer coefficients of its generators

  • From among the relevant parameters, we can highlight the ω-primality, the tame degree, the ∆-set and the elasticity. What these try to measure, in one way or another, is how far a semigroup or a ring is from having unique factorization, and if factorization is not unique, they explain its behaviour

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Summary

Introduction

A numerical semigroup (or numerical monoid) is a finitely generated subsemigroup of the set of nonnegative integers N, such that the group generated by it is the set of all integers Z. Every numerical semigroup is finitely generated and their elements might be expressed in different ways as a linear combination with non-negative integer coefficients of its generators Each such expression is usually known as a factorization of the element. If the ∆-set of an element is the empty set, this means that all its factorizations have the same length Computation of these parameters is not trivial, because, in general, their definitions might not be complicated, to establish appropriate and effective algorithms and relevant examples, it is necessary to have knowledge of a variety of properties (bounds, periodicity, etc.). }. The main goal of this work is to give properties of the set of lengths of a numerical semigroup and to obtain algorithms that allow computation of the function ∆ν.

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