A weaker notion of the finite factorization property

TL;DR

This paper explores a weaker form of the finite factorization property in positive monoids, identifying classes that satisfy it and comparing it to other known factorization properties to deepen understanding of non-unique factorizations.

Contribution

It introduces a new, weaker notion of the finite factorization property and characterizes classes of positive monoids satisfying this property, expanding the theoretical framework.

Findings

Identified large classes of positive monoids with the length-finite factorization property

Compared the length-finite property with bounded and finite factorization properties

Enhanced understanding of non-unique factorizations in positive monoids

Abstract

An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid $M$ is a length-finite factorization monoid if each $b \in M$ has only finitely many factorizations of any prescribed length. An additive submonoid of $\mathbb{R}_{\ge 0}$ is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years.