Elements with unique length factorization of a numerical semigroup generated by three consecutive numbers

TL;DR

This paper characterizes elements with unique length factorizations in a numerical semigroup generated by three consecutive integers, providing detailed descriptions and partitions based on length and denumerant, extending previous elementary results.

Contribution

It offers a detailed description of elements with uniform length factorizations in semigroups generated by three consecutive numbers, using Apéry sets and Betti elements to extend prior elementary findings.

Findings

Characterization of elements with all factorizations of the same length

Partitioning of these elements based on length and denumerant

Extension of previous elementary results using algebraic tools

Abstract

Let $S$ be the numerical semigroup generated by three consecutive numbers $a,a+1,a+2$, where $a\in\mathbb{N}$, $a\geq 3$. We describe the elements of $S$ whose factorizations have all the same length, as well as the set of factorizations of each of these elements. We give natural partitions of this subset of $S$ in terms of the length and the denumerant. By using Ap\'ery sets and Betti elements we are able to extend some results, first obtained by elementary means.